New Approach to Modeling Symbiosis in Biological and Social Systems
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International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
NEW APPROACH TO MODELING SYMBIOSIS INBIOLOGICAL AND SOCIAL SYSTEMS
V.I. YUKALOV
Department of Management, Technology and Economics,ETH Z¨urich, Swiss Federal Institute of Technology, Z¨urich CH-8092, SwitzerlandandBogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, [email protected]
E.P. YUKALOVA
Department of Management, Technology and Economics,ETH Z¨urich, Swiss Federal Institute of Technology, Z¨urich CH-8092, SwitzerlandandLaboratory of Information Technologies,Joint Institute for Nuclear Research, Dubna 141980, [email protected]
D. SORNETTE
Department of Management, Technology and Economics,ETH Z¨urich, Swiss Federal Institute of Technology, Z¨urich CH-8092, SwitzerlandandSwiss Finance Institute, c/o University of Geneva,40 blvd. Du Pont d’Arve, CH 1211 Geneva 4, [email protected]
Received (to be inserted by publisher)
We suggest a novel approach to treating symbiotic relations between biological species or socialentities. The main idea is the characterisation of symbiotic relations of coexisting species throughtheir mutual influence on their respective carrying capacities, taking into account that thisinfluence can be quite strong and requires a nonlinear functional framework. We distinguishthree variants of mutual influence, representing the main types of relations between species: (i)passive symbiosis, when the mutual carrying capacities are influenced by other species withouttheir direct interactions; (ii) active symbiosis, when the carrying capacities are transformedby interacting species; and (iii) mixed symbiosis, when the carrying capacity of one species isinfluenced by direct interactions, while that of the other species is not. The approach allows usto describe all kinds of symbiosis, mutualism, commensalism, and parasitism within a unifiedscheme. The case of two symbiotic species is analysed in detail, demonstrating several dynamicalregimes of coexistence, unbounded growth of both populations, growth of one and elimination ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette of the other population, convergence to evolutionary stable states, and everlasting populationoscillations. The change of the dynamical regimes occurs by varying the system parameterscharacterising the destruction or creation of the mutual carrying capacities. The regime changesare associated with several dynamical system bifurcations.
Keywords : Mathematical models of symbiosis, nonlinear differential equations, dynamics of coexistingspecies, functional carrying capacity, dynamical system bifurcations, supercritical Hopf bifurcationugust 4, 2014 0:17 symbiosis˙Subm˙ARX
New Approach to Modeling Symbiosis in Biological and Social Systems
1. Introduction different microorganisms.Other well known examples include the association between plant roots and fungi, between coral organismsand various types of algae, between cattle egrets and cattle, between alive beings and different bacteriaand viruses. Numerous other examples can be found in the literature [Boucher, 1988; Douglas, 1994; Sapp,1994; Ahmadjian & Paracer, 2000; Townsend et al., 2002; Yukalov et al., 2012a].Symbiotic relations are not restricted only to biological species, but may be noticed in a variety ofeconomic, financial, and other relations in social systems. For example, as a variant of symbiosis one cantreat the interconnections between economics and arts, between basic and applied science, between cultureand language, between banks and firms, between different enterprises, between firm owners and hiredpersonal, and so on. Many more examples are discussed in the following publications [Von Hippel, 1988;Grossman & Helpman, 1991; Richard, 1993; Graedel & Allenby, 2003; Yukalov et al., 2012a].Coexistence of different species, such as predators and preys, is usually described by the evolutionequations of the Lotka-Volterra type [Lotka, 1925], where the species directly interact with each other.Such equations are not valid for describing symbiosis since, in symbiotic relations, the species do noteat or kill each other, but influence their mutual livelihoods [Boucher, 1988; Douglas, 1994; Sapp, 1994;Ahmadjian & Paracer, 2000; Townsend et al., 2002; Yukalov et al., 2012a,b].From the mathematical point of view, the replicator equations characterizing trait groups, e.g., coop-erators, defectors, and punishers, are also of the Lotka-Volterra type, as they consider direct interactions.This also concerns the public-goods games for the spatially structured trait-group coexistence on networks[Perc & Szolnoki, 2010, 2012; Szolnoki & Perc, 2013a,b; Perc et al., 2013], as well as the utility-rate equa-tions [Yukalov et al., 2013]. Such equations do not seem to be applicable for describing symbiotic relations.The special feature of symbiotic relations is that the coexisting species interact with each othermainly through changing the carrying capacities of each other [Boucher, 1988; Douglas, 1994; Sapp, 1994;Ahmadjian & Paracer, 2000; Townsend et al., 2002; Yukalov et al., 2012a]. Therefore, for the correct de-scription of symbiosis, it is necessary to consider the species carrying capacities as functionals of the speciespopulation fractions. In the previous publications [Yukalov et al., 2012a,b], the species carrying capacityfunctionals were considered as polynomial expansions in powers of population fractions, with taking intoaccount only the lowest orders of these expansions, which resulted in linear or bilinear approximations,depending on the symbiosis type. Such lowest-order approximations presuppose that the symbiotic interac-tions are sufficiently weak, only slightly varying the mutual carrying capacities. The formal use of the linearor bilinear approximations in some cases leads to artificial finite-time singularity or finite-time populationdeath. This defect could be avoided by employing a nonlinear form for the carrying capacities.It is the aim of the present paper to formulate the evolution equations for coexisting symbiotic speciesinteracting through the influence on the carrying capacities of each other, keeping in mind that this influencecan be arbitrarily strong. This implies the use of a nonlinear form for the carrying capacity functional. Wefollow the idea used for the generalization of the evolution equation for a single population [Yukalov et al.,2014], where some artificial singularities could be eliminated by a nonlinear carrying-capacity functional.But now, we consider several symbiotic coexisting species, which makes the principal difference from thesingle-species evolution.In Sec. 2, we formulate the main idea of the approach for describing symbiosis, not as a direct interactionof different species as in the Lotka-Volterra equation but, through the mutual influence on the carryingcapacities of each other. We distinguish three types of the mutual influence characterizing passive symbiosis,active symbiosis, and mixed-type symbiosis. The origin of the problems spoiling the linear or bilinearapproximations is explained. In Sec. 3, we introduce the generalized formulation for the symbiotic equationswith nonlinear carrying-capacity functionals. We stress that this formulation allows one to treat symbioticrelations of arbitrary strength and of any type, such as mutualism, commensalism, and parasitism. In thefollowing sections, we give a detailed analysis of the dynamics and of the evolutionary stable states forugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette each kind of the symbiotic relations characterizing passive symbiosis (Sec. 4), active symbiosis (Sec. 5),and mixed symbiosis (Sec. 6) for the cases of mutualism and parasitism. All possible dynamical regimesare investigated, occurring as a result of bifurcations. Of special interest is a supercritical Hopf bifurcationhappening in the case of mixed symbiosis. Section 7 summarizes the results for commensalism that is amarginal case between mutualism and parasitism. Section 8 concludes.
2. Modeling Symbiosis by Mutual Influence on Carrying Capacities2.1.
General structure of equations
Suppose several species, enumerated as i = 1 , , . . . , S , coexist, being in mutual symbiosis with each other. Itis always admissible to reduce the consideration to dimensionless units, as has been thoroughly explainedin the previous articles [Yukalov et al., 2012a,b]. So, in what follows, we shall deal with dimensionlessquantities, such as the species fractions x i = x i ( t ) depending on dimensionless time t ≥
0. The speciescan proliferate, with the inverse of the proliferation rates characterizing their typical lifetimes. Gener-ally, in applications of dynamical theory, the time scales of the connected equations could be different[Desroches et al., 2012]. However, in the case of symbiosis, one has to keep in mind that the relationsbetween the species, by definition, are assumed to last sufficiently long to be treated as genuinely sym-biotic [Boucher, 1988; Douglas, 1994; Sapp, 1994; Ahmadjian & Paracer, 2000; Townsend et al., 2002;Yukalov et al., 2012a]. While this does not cover all symbiotic relationships, such as those between long-lived humans and very short-lived bacterias for instance, we will restrict our considerations to the class ofsymbiotic relationships in which the lifetimes of the involved species are comparable. This corresponds tothe situations where these species compete for the corresponding carrying capacities y i . Since the meaningof symbiotic relations is the variation of the mutual carrying capacities, the latter have to be functionalsof the population fractions: y i = y i ( x , x , . . . , x S ) . (1)We consider the society, composed of the symbiotic species, as being closed, so that the carrying capacitiesare not subject to variations caused by external forces, but are only influenced by the mutual interactionsof the coexisting species. Therefore the population dynamics is completely defined by the type of symbioticrelations between the species. In the other case, when external forces would be present, the society dynamicswould also be governed by such external forces and would strongly depend on the kind of chosen forces[Leonov & Kuznetsov, 2007; Pongvuthithum & Likasiri, 2010].The evolution equation for an i -th species takes the form dx i dt = x i − x i y i , (2)where x i and y i are non-negative. In particular, for the case of two species, x ≡ x and x ≡ z , we have dx ( t ) dt = x ( t ) − x ( t ) y ( x, z ) , dz ( t ) dt = z ( t ) − z ( t ) y ( x, z ) . (3)To proceed further, it is necessary to specify the expressions for y i . It is possible to distinguish three typesof symbiosis, depending on the process controlling the variations of the mutual carrying capacities. Passive weak symbiosis
To clearly characterize the difference between the symbiosis types, we start with the case of weak symbiosis,when the mutual influence on the carrying capacities is not strong, so that the carrying-capacity functionalscan be approximated by power-law expansions over the species fractions, limiting ourselves by the lowestorders of such expansions. In what follows, we consider the case of two symbiotic species.Probably the most often met type of symbiosis is when the species influence the mutual carryingcapacities just by the existence of the species themselves, resulting in the change on the mutual livelihoodscaused by the species vital activity. For instance, most land ecosystems rely on symbiosis between the plantsugust 4, 2014 0:17 symbiosis˙Subm˙ARX
New Approach to Modeling Symbiosis in Biological and Social Systems that extract carbon from the air and mycorrhizal fungi extracting minerals from the ground. If the mutualinfluence is weak, the corresponding carrying capacities can be modeled by the linear approximation y ( x, z ) ≃ b z , y ( x, z ) ≃ g x , (4)in which b and g are the parameters characterizing the productive or destructive influence of the relatedspecies on their counterparts. This type of symbiosis can be termed passive, since the species do not directlyinteract in the process of varying the livelihoods of their neighbors, in the sense that the impacts of speciesabundance on carrying capacities are linear. Active weak symbiosis
If in the process of influencing the livelihoods of each other the species directly interact, then their carryingcapacities are approximated by the bilinear expressions y ( x, z ) ≃ b xz , y ( x, z ) ≃ g xz , (5)assuming that their interactions are sufficiently weak. The parameters b and g again can be interpretedas production or destruction coefficients, according to the production or destruction affecting the relatedcarrying capacities. Examples of this type of symbiosis could be the relations between different firmsproducing goods in close collaboration with each other, or the relation between basic and applied sciences. Mixed weak symbiosis
The intermediate type of symbiosis is when one of the species, in the process of influencing the carryingcapacity, interacts with the other species, while the latter acts on the carrying capacity of the neighborjust by means of its vital activity, without explicit interactions. This type of symbiosis is common for manybiological and social systems, when one of them is a subsystem of the larger one. Thus, such relations existbetween the country gross domestic product and the level of science, or between culture and language. Inthe case of weak influence of this type, the carrying capacities are represented as the expansions y ( x, z ) ≃ b xz , y ( x, z ) ≃ g x . (6)
3. Arbitrarily Strong Influence on Livelihoods of Symbiotic Species
When one or both of the species are parasites, the corresponding parameters b i or g i are negative. Hencethe effective carrying capacity y i can become zero or negative. Then, with the approximations (4), (5) or(6), the evolution equations (3) can result in finite-time singularities of the population fraction or in thedisappearance of the solutions, as is found in the previous publications [Yukalov et al., 2012a,b]. Such asituation looks artificial, being just the result of the linear or bilinear approximations involved. In order toavoid that carrying capacities can become zero and negative, it is necessary to employ a more elaboratedexpression for the carrying-capacity functionals. Passive nonlinear symbiosis
In order to be able to consider mutual symbiotic relations of arbitrary strength, it is necessary to generalizethe carrying capacities. The latter can be treated as expansions in powers of the species fractions. In the caseof weak symbiosis, it has been possible to limit oneself to the first terms of such expansions. However, forgeneralizing the applicability of the approach, it is necessary to define the effective sums of such expansions.The effective summation of power series can be done by means of the self-similar approximation theory[Yukalov, 1990, 1991, 1992; Yukalov & Yukalova, 1996; Gluzman & Yukalov, 1997]. Under the condition ofobtaining an effective sum that does not change its sign for the variables defined on the whole real axis,one needs to resort to the exponential approximants [Yukalov & Gluzman, 1998; Gluzman & Yukalov,1998a,b,c; Yukalov & Gluzman, 1999; Gluzman et al., 2003]. Limiting ourselves by the simplest form ofsuch an exponential approximation, for the case of expansion (4), we have the effective sums y ( x, z ) = e bz , y ( x, z ) = e gx . (7)ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette
The parameters b and g , depending on their signs, characterize the creative or destructive influence of thespecies on the carrying capacities of their coexisting neighbors. Active nonlinear symbiosis
Similarly involving the method of self-similar exponential approximants to perform the summation for theprevious case of passive symbiosis, starting from the terms (5), we obtain the nonlinear carrying capacities y ( x, z ) = e bxz , y ( x, z ) = e gxz , (8)with the same interpretation of the symbiotic parameters b and g . Mixed nonlinear symbiosis
In the case of the mixed symbiosis, with the carrying capacities starting from terms (6), we get the effectivesummation resulting in the nonlinear expressions y ( x, z ) = e bxz , y ( x, z ) = e gx . (9)In this way, we obtain the generalizations for the carrying capacities that are applicable to the mutualsymbiotic relations of arbitrary strength. Variants of symbiotic coexistence
It is worth stressing that the suggested approach allows us to treat all known variants of symbiosis, whichis regulated by the values of the symbiotic parameters b and g . Thus, when both these parameters arepositive, this corresponds to mutualism, which is a relationship between different species where both ofthem derive mutual benefit: b > , g > mutualism ) . (10)When one of the species benefits from the coexistence with the other species, while the other one isneutral, getting neither profit nor harm, this relationship corresponds to commensalism, defined by one ofthe conditions b > , g = 0 ; b = 0 , g > commensalism ) . (11)Finally, if at least one of the coexisting species is harmful to the other one, this is typical of parasitismthat is characterized by the validity of one of the conditions b > , g < b = 0 , g < b < , g < b < , g = 0 ; b < , g > parasitism ) . (12)Varying the symbiotic parameters b and g results in a variety of bifurcations between different dy-namical regimes. Below, we study these bifurcations employing the general methods of dynamical theory[Thompson et al., 1994; Kuznetsov, 1995; Chen et al., 2003; Leonov & Kuznetsov, 2013].
4. Passive Nonlinear Symbiosis
Equations (3), with the carrying capacities (7) acquire the form dxdt = x − x e − bz , dzdt = z − z e − gx , (13)with the parameters g ∈ ( −∞ , ∞ ) and b ∈ ( −∞ , ∞ ), and with the initial conditions x = x (0), z = z (0).We are looking for non-negative solutions x ( t ) ≥ z ( t ) ≥ New Approach to Modeling Symbiosis in Biological and Social Systems We may note that system (13) is symmetric with respect to the change b ←→ g , x ←→ z , and x ( t ) ←→ z ( t ).The overall phase portrait for the case of the passive symbiosis will be displayed below. Existence of evolutionary stable states
Depending on the signs of the parameters g and b , the system of equations (13) possesses different numbersof stationary states (fixed points) { x ∗ , z ∗ } . For any values of the parameters, there always exist three trivialfixed points, { x ∗ = 0 , z ∗ = 0 } , { x ∗ = 1 , z ∗ = 0 } , and { x ∗ = 0 , z ∗ = 1 } , which are unstable for all g and b .Nontrivial fixed points { x ∗ = 0 , z ∗ = 0 } are the solutions to the equations: x ∗ = e bz ∗ , z ∗ = e gx ∗ . (14)The characteristic exponents, defining the stability of the stationary solutions, are given by the ex-pression λ , = − ± p bgx ∗ z ∗ , (15)where x ∗ and z ∗ are the fixed points defined by Eqs. (14), which can also be represented as x ∗ = exp( be gx ∗ ) , z ∗ = exp( ge bz ∗ ) . Analysing the existence and stability of the fixed points, we find that, when b ∈ (0 , ∞ ) and g ∈ (0 , ∞ ),there exists a line g = g c ( b ), with respect to which two possibilities can occur: • If 0 < g < g c ( b ), then Eqs. (14) have two solutions { x ∗ , z ∗ } and { x ∗ , z ∗ } , such that 1 < x ∗ < x ∗ and1 < z ∗ < z ∗ . Numerical investigation of (14) and (15) shows that the lower fixed point { x ∗ , z ∗ } is stable,while the higher fixed point { x ∗ , z ∗ } is unstable. • If g > g c ( b ), then Eqs. (14) do not have solutions, hence Eqs. (13) do not possess stationary states. When b = 1 /e , then g c ( b ) = 1 /e and x ∗ = z ∗ = e .If either b ≤ g >
0, or b > g ≤
0, then there exists only one stationary solution { x ∗ , z ∗ } , which is a stable fixed point characterized by the corresponding characteistic exponents (15) withRe λ , <
0, which follows from the inequalities λ λ = 1 − bgx ∗ z ∗ > − ( λ + λ ) = 2 > b ≤ g >
0, then x ∗ < z ∗ >
1, while if b > g ≤
0, then x ∗ > z ∗ < b < g <
0, there can exist up to three fixed points, such that x ∗ < z ∗ <
1. For b ≤ − e , there exist two lines g = g ( b ) < g = g ( b ) <
0, for which g ( − e ) = g ( − e ) = − e . Thefollowing two possibilities can happen: • If b < − e and g ( b ) < g < g ( b ), then Eqs. (14) have 3 solutions. Numerical analysis shows that two fixedpoints, { x ∗ , z ∗ } and { x ∗ , z ∗ } , are stable, while the third fixed point, { x ∗ , z ∗ } , is unstable. • If either − e ≤ b < g <
0, or b < − e and g < g ( b ), or b < − e and g ( b ) < g <
0, then Eqs. (14) haveonly one solution, which is a stable fixed point.In the limiting cases, if g →
0, then z ∗ → x ∗ → e b . If b →
0, then x ∗ → z ∗ → e g .If b → −∞ and g > x ∗ → z ∗ →
1. When g → −∞ and b > z ∗ → x ∗ → b ≪ −
1, and g ≪ −
1, then, depending on the relation between the parameters, it may be that either x ∗ → z ∗ →
1, or x ∗ → z ∗ →
0, or x ∗ → z ∗ →
0, but at finite parameter values, onealways has finite x ∗ = 0 and z ∗ = 0.Figure 1a clarifies the regions of the stable fixed-point existence in the plane b − g . The region ofexistence for b, g < Dynamics of populations under mutualism ( b > , g > ) There exist two types of dynamical behavior: (i)
Unbounded growth of populations or (ii) convergence to astationary state .ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette (i)
Unbounded growth of populations
When 0 < b < ∞ and g > g c ( b ) >
0, then Eqs. (14) do not have solutions, hence Eqs. (13) do not havestationary states. For any choice of the initial conditions { x , z } , solutions x ( t ) , z ( t ) → ∞ , as t → ∞ . Thecorresponding behavior of x ( t ) and z ( t ) is shown in Fig.2a.(ii) Convergence to stationary states
If 0 < b < ∞ and 0 < g < g c ( b ), there exist two solutions to Eqs. (14), such that 1 < x ∗ < x ∗ and1 < z ∗ < z ∗ . Numerical analysis shows that the lower solution { x ∗ , z ∗ } is stable, while { x ∗ , z ∗ } is unstable.Around the stable fixed point, there exists a finite basin of attraction, so that if the initial conditions { x , z } belong to it, then x ( t ) → x ∗ , and z ( t ) → z ∗ , as t → ∞ . But if the initial conditions do not belongto the basin of attraction, the species fractions rise to infinity for t → ∞ .Figure 2b demonstrates the behavior or solutions for g slightly above the boundary ( g > g c ( b )) andslightly below it ( g < g c ( b )). Below the boundary g c = g c ( b ), there exists a region of parameters b, g , suchthat x ( t ) and z ( t ) converge to a stationary state, provided that the initial conditions are inside the basin ofattraction. Figures 2c and 2d illustrate the behaviour of x ( t ) and z ( t ) for the same choice of the parameters b, g , but for different initial conditions. Dynamics of populations with one parasitic species (either b < , g > or b > , g < ) Only one regime exists, when the populations tend to their stationary states , with the attraction basinbeing the whole region of x , z .Recall that there is a symmetry in Eqs. (13), such that under the replacement b ←→ g and x ←→ z ,we have x ( t ) ←→ z ( t ) and x ∗ ←→ z ∗ . Therefore, it is sufficient to consider only one case, say, when b < g > b < g >
0, Eqs. (14) have a single solution { x ∗ < , z ∗ > } , which is a stable fixed point.Conversely, when b > g <
0, then x ∗ > z ∗ <
1. This tells us that the species populationcoexisting with a parasite is suppressed.The populations converge to their stationary points irrespectively of the choice of the initial conditions.The convergence can be either monotonic or non-monotonic, as is demonstrated in Figs. 3 and 4.
Dynamics of populations with two parasitic species ( b < , g < ) Depending on the symbiotic parameters, there can exist either (i) a single stationary state or (ii) bistabilitywith two stationary states .When b < g <
0, then Eqs. (14) have either a single solution { x ∗ , z ∗ } , which is a stable fixedpoint, or three solutions, among which two solutions, { x ∗ , , z ∗ , } , are stable fixed points and the third, { x ∗ , z ∗ } , is a saddle. In all the cases, x ∗ < z ∗ <
1, which means that two parasitic species cannotdevelop large populations.(i)
Single stationary state
For b, g ≤ − e , there exist two lines, g ( b ) and g ( b ), such that g ( − e ) = g ( − e ) = − e and g ( b ) 0. If either − e < b < g < 0, or b < − e and g ( b ) < g < 0, or b < − e and g < g ( b ), thenthere exists a single solution to Eqs. (14), which is a stable fixed point.(ii) Bistability with two stationary states .If b < − e and g ( b ) < g < g ( b ), there exist three solutions to Eqs. (14). Two solutions are stable fixedpoints and the third solution is a saddle. The bifurcation point corresponds to b = g = − e . At this point, x ∗ = z ∗ = 1 /e .The populations x ( t ) and z ( t ) tend monotonically or non-monotonically to their stable stationarystates, from above or from below, as is shown in Figs. 5 and 6. In the region of the symbiotic parameters b, g < 0, where a single stable fixed point exists, x ( t ) → x ∗ , z ( t ) → z ∗ , as t → ∞ , irrespectively of theinitial conditions.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX New Approach to Modeling Symbiosis in Biological and Social Systems In the region of the parameters b and g , where two stable fixed point exist, the population convergencedepends on the choice of initial conditions. For t → ∞ , the population { x ( t ) , z ( t ) } tend either to { x ∗ , z ∗ } or to { x ∗ , z ∗ } , depending on the initial conditions { x , z } being in the attraction basin of the related fixedpoint.On the line − e < b = g < 0, there exists a single stationary state { x ∗ = z ∗ } that is a stable fixed point.On the line b = g < − e , there are three fixed points. Two fixed points are stable, so that x ∗ = z ∗ and z ∗ = x ∗ . The third fixed point, { x ∗ = z ∗ } is unstable. For b = g → −∞ , we have x ∗ → z ∗ → x ∗ → z ∗ → 0, and x ∗ = z ∗ → Phase portrait for passive nonlinear symbiosis Fig. 7 provides an overview of the different regimes analysed in this section concerned with the analysis ofequation (13).Panels (a) and (b) illustrate the regime of mutualism, in which the two species benefit from each other.Panel (a) represents the situation of sufficiently symmetric mutualism, in which the long-term behaviouris characterised by an exponential growth for both species. Note that asymmetric initial conditions leadto a non-monotonous behaviour of the species population that is initially too large, which has to shrinkfirst before growing again in synergy with the other species. Panel (b) corresponds to the situation of largemutualism asymmetries. In this case, the two population need to be above a certain threshold in order toreach the regime of unbounded growth. Otherwise, their populations are trapped and converge to a steadystate.Panel (c) illustrates the dynamics of a very asymmetric situation where one species augments thecarrying capacity of the other, while the later has a negative effect on the carrying capacity of the former.In this case, the dynamics of the two species population converges to a stable fixed point, a steady statecharacterising a compromise in the destructive interactions between the two species.In the situation shown in panel (d) in which both species tend to destroy the carrying capacity of theother, a saddle node separates two symmetrical stable fixed points, where one species in general profitsmuch more than the other. This is an example of a spontaneous symmetry breaking, in which the twostable fixed points are exactly symmetric under the transformation x ←→ z . The selection of one or theother stable fixed points depends on the initial conditions, as usual in spontaneous symmetry breaking.A slight initial advantage of x above z is sufficient to consolidate into a very large asymptotic populationdifference. Such behaviours are reminiscent to many real-life situations in which a slight initial favourablesituation becomes entrenched in a very strong prominent role. Our analysis shows that such a situationoccurs generically in the case of passive symbiosis when the two species are competing destructively. It istempting to interpret real-life geopolitical and economic situations in particular as embodiment of such ascenario. 5. Active Nonlinear Symbiosis This case is described by the system of equations dxdt = x − x e − bxz , dzdt = z − z e − gxz , (16)with the symbiotic parameters g ∈ ( −∞ , ∞ ), b ∈ ( −∞ , ∞ ), and the initial conditions x = x (0), z = z (0).Again, only non-negative solutions x ( t ) ≥ z ( t ) ≥ b ←→ g , x ←→ z , and x ( t ) ←→ z ( t ). Existence of evolutionary stable states Similarly to the previous case, there always exist three trivial fixed points, { x ∗ = 0 , z ∗ = 0 } , { x ∗ = 1 , z ∗ =0 } , and { x ∗ = 0 , z ∗ = 1 } , which are unstable for any g and b .Nontrivial fixed points { x ∗ = 0 , z ∗ = 0 } are the solutions to the equations: x ∗ = e bx ∗ z ∗ , z ∗ = e gx ∗ z ∗ . (17)ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette These Eqs. (17) can be represented as x ∗ = exp( b ( x ∗ ) g/b ) , z ∗ = exp( g ( z ∗ ) b/g ) . (18)The characteristic exponents are given by the equations λ = − , λ = − b + g ) x ∗ z ∗ , (19)where x ∗ and z ∗ are the solutions to Eqs. (17). From here, it follows that the fixed points are stable for all b + g < • If b + g > /e , then Eqs. (17), or (18), do not have solutions. • If 0 < b + g < /e , then Eqs. (17) or (18) have two solutions { x ∗ , z ∗ } and { x ∗ , z ∗ } .Here, when b > g > 0, then 1 < x ∗ < x ∗ and 1 < z ∗ < z ∗ . When b < g > 0, then 1 > x ∗ > x ∗ and1 < z ∗ < z ∗ . If b > g < 0, then 1 < x ∗ < x ∗ , but 1 > z ∗ > z ∗ . Numerical investigation shows thatthe fixed point { x ∗ , z ∗ } is stable, while the second fixed point { x ∗ , z ∗ } is a saddle. • If b + g < 0, then there exists only one solution to Eqs. (17) or (18) that is a stable fixed point.The region of the existence of the stable fixed-point in the plane b − g is presented in Fig. 8.When 0 < b + g < /e , the stable fixed point possesses a finite basin of attraction. If b + g ≤ 0, thenthe basin of attraction is the whole region { x ≥ , z ≥ } . Population dynamics under b + g > /e In this case, Eqs. (17) or (18) do not have solutions, which means that there are no fixed points. There canhappen two types of solutions: (i) Unbounded growth of mutualistic populations or (ii) Unbounded growthof a parasitic population and dying out of the host population .(i) Unbounded growth of mutualistic populations In the case of mutualism, when b > g > 0, both populations grow so that x ( t ) → ∞ and z ( t ) → ∞ , as t → ∞ .(ii) Unbounded growth of parasitic population and dying out of host population When one of the species is parasitic, the host population dies out. If b < 0, but g > 0, then x ( t ) → z ( t ) → ∞ , as t → ∞ . Conversely, if b > 0, but g < 0, then x ( t ) → ∞ , while z ( t ) → 0, as t → ∞ .The population dynamics for the case b + g > /e is presented in Fig. 9. Because of the symmetry ofEqs. (16), it is sufficient to consider only the cases when b > g > b < g > Population dynamics under b + g < /e There exists only one stable fixed point, so that the population dynamics depends on whether the initialconditions are inside the attraction basin or not. Respectively, there can happen three kinds of behavior:(i) Convergence to a stationary state ; (ii) Unbounded growth of both populations ; (iii) Unbounded growth ofparasitic population and dying out of host population .(i) Convergence to stationary state When 0 < b + g < /e and the initial conditions { x , z } are in the basin of attraction of the fixedpoint { x ∗ , z ∗ } , then the populations x ( t ) → x ∗ and z ( t ) → z ∗ , as t → ∞ .If b + g ≤ 0, then the basin of attraction is the whole region of x ≥ z ≥ 0, that is, for anychoice of the initial conditions { x , z } the solutions { x ( t ) , z ( t ) } → { x ∗ , z ∗ } , as t → ∞ .(ii) Unbounded population growth When 0 < b + g < /e and b, g > 0, there exists a finite basin of attraction, so that if the initialconditions { x , z } are outside of the attraction basin, then the populations { x ( t ) , z ( t ) } unboundedly growwith increasing time t → ∞ .ugust 4, 2014 0:17 symbiosis˙Subm˙ARX New Approach to Modeling Symbiosis in Biological and Social Systems (iii) Unbounded growth of parasitic population and dying out of host population When 0 < b + g < /e , with b < g > 0, and initial conditions { x , z } are outside of the attractionbasin, then for t → ∞ , one of the species is dying out, x ( t ) → 0, while another one experiences unboundedgrowth z ( t ) → ∞ . Conversely, if b > 0, but g < 0, then x ( t ) → ∞ , while z ( t ) → 0, as t → ∞ .Figure 10 demonstrates different types of the population convergence to a stable stationary state { x ∗ , z ∗ } . Phase portrait for active nonlinear symbiosis Figure 11 provides an overview of the different regimes analysed in this section concerned with the analysisof equation (16). Qualitatively, the nature of the different regimes are similar to those discussed with respectto the phase portrait shown in Fig. 7 summarising the different regime for passive symbiosis described byequation (13). 6. Mixed Nonlinear Symbiosis The system of equations describing this case is dxdt = x − x e − bxz , dzdt = z − z e − gx , (20)with the symbiotic parameters g ∈ ( −∞ , ∞ ), b ∈ ( −∞ , ∞ ), and the initial conditions x = x (0), z = z (0).As usual, only non-negative solutions x ( t ) ≥ z ( t ) ≥ x ( t ) is formed in the process of interactions between the coexisting species. Incontrast, the carrying capacity of the population z ( t ) is influenced only by the population x ( t ). Therefore,we may call the population x ( t ) active , but the population z ( t ) passive . Existence of evolutionary stable states Similarly to the previous cases, there always exist 3 trivial fixed points, { x ∗ = 0 , z ∗ = 0 } , { x ∗ = 1 , z ∗ = 0 } ,and { x ∗ = 0 , z ∗ = 1 } , which are unstable for all b, g ∈ ( −∞ , ∞ ).Nontrivial fixed points { x ∗ = 0 , z ∗ = 0 } are the solutions to the equations: x ∗ = e bx ∗ z ∗ , z ∗ = e gx ∗ , (21)which can be transformed into x ∗ = exp( bx ∗ e gx ∗ ) , z ∗ = exp( gz bz ∗ /g ) . (22)The characteristic exponents, defining the stability of the stationary states, are given by the equations λ , = 12 h bx ∗ z ∗ − ± x ∗ p bz ∗ (4 g + bz ∗ ) i , (23)The following cases can occur: • When b > 0, there exists a boundary g = g c ( b ), such that if g < g c ( b ), then there can exist up to threefixed points, but only one of them being stable. • If either 0 < b < /e and g > g c ( b ) > 0, or b > /e and g ≥ 0, then Eqs. (21) do not have solutions, hence,there are no fixed points. • For b > /e and g c ( b ) < g < 0, there exists only one unstable fixed point. • If 0 < b < /e and 0 ≤ g < g c ( b ), there is one stable and one unstable fixed points. • When b ≤ g ∈ ( −∞ , ∞ ), there exists a single fixed point that is stable. For b < g > 0, we have x ∗ < , z ∗ > 1, while when b < g < 0, then x ∗ < , z ∗ < • For 0 ≤ b ≤ b ≈ . 47, there exists an additional line g ( b ), such that g (0) = 0 and g ( b ) = g c ( b ) ≈− . < b < b and g < g ( b ) or b ≥ b and g < g c ( b ), there is a single fixed point that isstable.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette • When either 0 < b < /e and g ( b ) < g < 0, or 1 /e ≤ b < b and g ( b ) < g < g c ( b ), then Eqs. (21) havethree solutions, but only one of them is a stable fixed point.In the limiting case, we have g c (1 /e ) = 0. When b → +0, then g c ( b ) → ∞ . If b → ∞ , then g c ( b ) → −∞ .On the boundary { b ∈ (0 , b ) , g = g ( b ) } , two solutions, corresponding to unstable fixed points, coincideand disappear as soon as g ≤ g ( b ).The region of the existence of the fixed-point in the plane b − g is presented in Fig. 12.The dynamics of the symbiotic populations strongly depends on whether the influence of the passivespecies z ( t ) on the active species x ( t ) is mutualistic or parasitic, which is described by their interactionsymbiotic parameter b . Therefore, these two cases will be treated separately. Dynamics of populations with mutualistic passive species ( b > ) Depending on the symbiotic parameters b and g , there are three types of dynamic behavior: (i) Unboundedgrowth of populations ; (ii) everlasting oscillations , and (iii) convergence to a stationary state .(i) Unbounded growth of populations When either 0 < b < /e and g > g c ( b ) > 0, or b > /e and g ≥ 0, then Eqs. (20) do not have fixedpoints. Populations x ( t ) → ∞ and z ( t ) → ∞ , either monotonically or non-monotonically, as t → ∞ . Thelogarithmic behaviour of diverging solutions x ( t ) and z ( t ) is shown in Figs. 13a and 13b.When 0 < b < /e and 0 ≤ g < g c ( b ), there are two fixed points, but only one of them is stable. Inthis case, there exists a finite basin of attraction for the stable fixed point. If the initial conditions { x , z } are outside of the attraction basin, the populations x ( t ) and z ( t ) diverge, as t → ∞ .(ii) Everlasting oscillations When b > /e and g c ( b ) < g < 0, then Eqs. (21) have only one solution { x ∗ , z ∗ } , which is anunstable focus, with complex characteristic exponents, such that λ = λ ∗ and Re λ , > 0. In this regionof the parameters b and g , there exists a limit cycle, and the populations x ( t ) and z ( t ) oscillate withoutconvergence for all t > g → − 0, then the amplitude of oscillations drastically increases, but remains finite. On the halfline, b > /e and g = 0, solutions x ( t ) and z ( t ) diverge, as t → ∞ .The dynamics of populations x ( t ) and z ( t ) are shown in Figs. 13c,d, 14a,b, 15c,d, and 16c,d. Figure17 presents the logarithmic behavior of the populations x ( t ), z ( t ) for the fixed b = 0 . g , suchthat either g c ( b ) < g < g ( b ) < g < g c ( b ).(iii) Convergence to stationary states When b > g < g c ( b ), there exists one stable fixed point { x ∗ , z ∗ } .When 0 < b < /e and 0 < g < g c ( b ), then there exists a finite basin of attraction for the fixed point.If the initial conditions { x , z } are inside the basin of attraction, then x ( t ) → x ∗ and z ( t ) → z ∗ , as t → ∞ .If either 0 < b < /e and g ≤ 0, or b > /e and g < g c ( b ) < 0, then the basin of attraction is the wholeregion of { x ≥ , z ≥ } . In that case, for any initial conditions, x ( t ) → x ∗ and z ( t ) → z ∗ , as t → ∞ .There are two possible ways of convergence to a stationary state, either with oscillations, when thefixed point is a focus, or without oscillations, when the fixed point is a node.When b > b and g < g c ( b ), and g is in the vicinity of g = g c ( b ), then x ( t ) → x ∗ and z ( t ) → z ∗ with oscillations. The corresponding behaviour is shown in Figs. 14c,d, 15c,d (solid lines), and Figs. 16c,d(dashed lines).When 0 < b < /e and 0 < g < g c ( b ) and the initial conditions { x , z } are inside the attraction basinof the fixed point, then x ( t ) → x ∗ , z ( t ) → z ∗ without explicit oscillations, though the convergence can beeither monotonic or non-monotonic.If either 0 < b < /e and g ≤ 0, or 1 /e < b < b and g < g c ( b ), or b > b and g ≪ g c ( b ), then x ( t ) → x ∗ and z ( t ) → z ∗ without oscillations.Examples of convergence without oscillations are given in Figs. 15 and 16.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX New Approach to Modeling Symbiosis in Biological and Social Systems The bifurcation line g = g c ( b ) < b > b is the line of a supercritical Hopf bifurcation. It separatesthe regions where the real parts of the complex characteristic exponents have different signs. When b > b and g < g c ( b ) < 0, then the characteristic exponents of the stable fixed point are complex, with thenegative Lyapunov exponents Re λ , < 0. When b > b and g c ( b ) < g < 0, then the stable focus transformsinto an unstable focus, with positive Lyapunov exponents Re λ , > 0. At the same time, there appearsa stable limit cycle. On the bifurcation line, where b > b and g = g c ( b ) the Lyapunov exponents of thefocus are zero, Re λ , = 0, as it should be under a Hopf bifurcation [Kuznetsov, 1995; Chen et al., 2003;Cobiaga & Reartes, 2013].A very interesting behaviour is associated with the transition across the line g c ( b ), when 1 /e < b < b .In the region where either 0 < b < /e and g < g < 0, or 1 /e < b < b and g ( b ) < g < g c ( b ), there arethree fixed points, a stable node, a saddle, and an unstable focus. In the region b > /e and g > g c ( b ), thereis one unstable focus and a limit cycle. When approaching the boundary g c ( b ), moving from one region tothe other, the stable node and saddle become close to each other and on the boundary they coincide, whiledisappearing to the right from this line g c ( b ). The unstable focus safely moves through the boundary tothe region where a limit cycle appears. Dynamics of populations with parasitic passive species ( b < ) There is only one regime of convergence to stationary states .When b ≤ g ∈ ( −∞ , ∞ ), there exists just one solution to Eqs. (21), or (22), and it is a stablefixed point. The attraction basin is the whole region of the initial conditions { x , z } , so that, for any initialcondition, populations x ( t ) → x ∗ and z ( t ) → z ∗ , as t → ∞ .The corresponding behaviour of the populations is presented in Fig. 16. Phase portrait for mixed nonlinear symbiosis The phase portraits for the case of the mixed symbiosis described by equation (20), under different symbioticparameters, are presented in Figs. 18 and 19.Fig. 18 for g > b > g < 0, so that the species x tends to destroythe carrying capacity of the species z , while the latter tends to augment nonlinearly the carrying capacityof the former. As a consequence of the asymmetry of the equations and of this pair of parameters, eitherconvergent oscillations to a stable fixed point or oscillatory solutions occur. 7. Commensalism as Marginal Type of Symbiosis (either b = 0 , or g = 0 )7.1. Independent species ( b = 0 and g = 0 ) This is the extreme case, when the species are actually independent of each other. If b = 0 and g = 0, thenthe systems of the evolution equations turn into two independent logistic equations dxdt = x − x , dzdt = z − z , (24)with the initial conditions x = x (0) and z = z (0).Solutions to these equations are known: x ( t ) = x x − ( x − e − t , z ( t ) = z z − ( z − e − t . (25)That is, x ( t ) → x ∗ = 1 and z ( t ) → z ∗ = 1, as t → ∞ .ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette Commensalism under b = 0 , g = 0 When b = 0 and g = 0 in equation (20), this can be treated as a limiting case of either passive or mixedsymbiosis. Then the system of equations (13), or (20), turns into dxdt = x − x , dzdt = z − z e − gx ( t ) . (26)The solution x ( t ) to the first equation of (26) is given by the first equation in (25). The stable set of fixedpoints is defined by the expressions x ∗ = 1 , z ∗ = e g , which exist for all g ∈ ( −∞ , ∞ ). Populations { x ( t ) , z ( t ) } → { x ∗ , z ∗ } monotonically either from above orfrom below, depending on the given initial conditions. Commensalism under b = 0 , g = 0 When b = 0 and g = 0 in equation (20), this is the limiting case of either active or mixed symbiosis. Thesystem of equations (16), or (20), becomes dxdt = x − x e − bxz , dzdt = z − z . (27)The solution z ( t ) to the second equation of (27) is given by the second expression in (25). The fixed point z ∗ = 1 is stable, while the fixed point z ∗ = 0 is unstable.For the first equation in (27), the trivial fixed point x ∗ = 0 is also unstable. Nontrivial fixed points x ∗ = 0 are the solutions to the equation x ∗ = e bx ∗ . (28)When b ≤ x ∗ ≤ x ( t ) → x ∗ and z ( t ) → z ∗ = 1, as t → ∞ .If 0 < b < /e , there exist two solutions, but only one solution, such that 1 < x ∗ < e , is stable,possessing a finite basin of attraction.When b > /e , there are no fixed points for x ( t ), hence x ( t ) → ∞ , though z ( t ) → 1, as t → ∞ .Dynamics of the populations x ( t ) and z ( t ) in the case of commensalism, described by Eqs. (24), (26),and (27), is demonstrated in Fig. 20. 8. Summary of Main Symbiotic Behaviors For the convenience of the reader, we summarize here the main types of symbiotic behaviors for theconsidered cases of symbiosis, classifying the population dynamics with respect to the related symbioticparameters. Dynamics under passive symbiosis The following types of population behavior can happen: • Unbounded growth for 0 < b < ∞ , g > g c ( b ) > . (29) • Convergence to stationary states , if the initial conditions are inside the attraction basin, and unboundedgrowth , when the initial conditions are outside the attraction basin, which occurs for0 < b < ∞ , < g < g c ( b ) . (30)ugust 4, 2014 0:17 symbiosis˙Subm˙ARX New Approach to Modeling Symbiosis in Biological and Social Systems • Convergence to stationary states for arbitrary initial conditions, when either b < , g > , (31)or b > , g < . (32) • Convergence to stationary states in the presence of bistability , when b < , g < . (33) Dynamics under active symbiosis The following regimes of population dynamics exist: • Unbounded growth if either b + g > e , b > , g > , (34)or b + g < e , b > , g > , (35)and initial conditions are outside of the attraction basin. • Dying out of the host species and unbounded proliferation of parasitic species , when either b < , g > e + | b | , (36)or b > e + | g | , g < , (37)or b < , | b | < g < e + | b | , (38)and initial conditions are outside of the attraction basin, or g < , | g | < b < e + | g | , (39)and initial conditions are outside of the attraction basin. • Convergence to stationary states for the initial conditions inside the attraction basin if b + g ≤ e . (40) Dynamics under mixed symbiosis There can exist the following regimes of population dynamics: • Unbounded growth when either 0 < b < e , g > g c ( b ) > , (41)or b > e , g ≥ . (42) • Convergence to stationary states for the initial conditions inside the attraction basin and unbounded growth for the initial conditions outside the attraction basin, when0 < b < e , ≤ g < g c ( b ) . (43)ugust 4, 2014 0:17 symbiosis˙Subm˙ARX V.I. Yukalov, E.P. Yukalova, D. Sornette • Convergence to stationary states for all initial conditions, if either b > e , g < g c ( b ) < , (44)or 0 < b < e , g ≤ , (45)or b ≤ , −∞ < g < ∞ . (46) • Everlasting oscillations under the condition b > e , g c ( b ) < g < . (47) 9. Conclusion A novel approach to treating symbiotic relations between biological or social species has been suggested.The principal idea of the approach is the characterization of symbiotic relations of coexisting speciesthrough the carrying capacities of each other. Taking into account that the mutual influence can be quitestrong, the carrying capacities are modeled by nonlinear functionals. We opt for the exponential form ofthis functional, since such a form naturally appears as an effective sum of a series, summed in a self-similarway, under the condition of semi-positivity.We distinguish three variants of mutual influence, depending on the type of relations between thespecies. In the case of passive symbiosis , the mutual carrying capacities are influenced by other specieswithout their direct interactions. In active symbiosis , the carrying capacities are influenced by interactingspecies. And mixed symbiosis describes the situation when the carrying capacity of one species is influencedby direct interactions, while that of the other species is not.The approach allows us to describe all kinds of symbiosis, that is, mutualism, commensalism, andparasitism . The case of two symbiotic species is analyzed in detail. Depending on the symbiotic parameters,characterizing the destruction or creation of the mutual carrying capacities, that is, describing mutualisticor parasitic species, several dynamical regimes of coexistence are possible: unbounded growth of bothpopulations, growth of one and the elimination of the other population, convergence to evolutionary stablestates, and everlasting population oscillations.The main difference of the described symbiosis with nonlinear functional carrying capacities, comparedwith the model of [Yukalov et al., 2012a,b] considered earlier of symbiosis with carrying capacities in thelinear or bilinear approximations, is the absence of finite-time singularities and of an abrupt death ofpopulations. The latter have appeared in the previous works [Yukalov et al., 2012a,b] owing to the artificialchange of sign of the carrying capacities, when they were crossing zero. The nonlinear carrying capacities,employed in the present paper, are semi-positive defined, which ensures that the crossing-zero problem isavoided.Among various bifurcations, when the dynamic regimes of the population evolution qualitativelychanges, the most interesting ones are the two bifurcations occurring for the case of mixed symbiosis,with the coexistence of two parasitic species, when crossing the bifurcation line g c ( b ). One is the super-critical Hopf bifurcation, when a stable focus becomes unstable and a limit cycle arises. The other is thecoalescence of a stable node with a saddle, with their disappearance and the appearance of a limit cycle.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES References Ahmadjian, V. & Paracer, S. [2000] An Introduction to Bilogical Associations (Oxford University, Oxford).Boucher, D. [1988] The Biology of Mutualism: Ecology and Evolution (Oxford University, New York).Chen, G., Hill, D.J. & Yu, X.H. [2003] Bifurcation Control: Theory and Applications (Springer, Berlin).Cobiaga, R, $ Reartes, W. [2013] “A new approach in the search for periodic orbits”, Int. J. Bifur. Chaos , 1350186.Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H. M. & Wechselberger, M. [2012]“Mixed-mode oscillations with multiple time scales”, SIAM Review , 211–288.Douglas, A. E. [1994] Symbiotic Interactions (Oxford University, Oxford).Gluzman, S. & Yukalov, V. I. [1997] “Algebraic self-similar renormalization in the theory of critical phe-nomena”, Phys. Rev. E , 3983–3999.Gluzman, S. & Yukalov, V. I. [1998a] “Resummation method for analyzing time series”, Mod. Phys. Lett.B , 61–74.Gluzman, S. & Yukalov, V. I. [1998b] “Renormalization group analysis of October market crashes”, Mod.Phys. Lett. B , 75–84.Gluzman, S. & Yukalov, V. I. [1998c] “Booms and crashes in self-similar markets”, Mod. Phys. Lett. B ,575–587.Gluzman, S., Sornette, D. & Yukalov, V. I. 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[2002] Ecology: Individuals, Populations and Communities (Blackwell Science, Oxford).Von Hippel, E. [1988] The Sources of Innovation (Oxford University, Oxford).Yukalov, V. I. [1990] “Self-similar approximations for strongly interacting systems”, Physica A , 833–860.Yukalov, V. I. [1991] “Method of self-similar approximations”, J. Math. Phys. , 1235–1239.Yukalov, V. I. [1992] “Stability conditions for method of self-similar approximations”, J. Math. Phys. ,3994–4001.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES Yukalov, V. I. & Yukalova, E.P. [1996] “Temporal dynamics in perturbation theory”, Physica A ,336–362.Yukalov, V. I. & Gluzman, S. [1998] “Self-similar exponential approximants”, Phys. Rev. E , 1359–1382.Yukalov, V. I. & Gluzman, S. [1999] “Weighted fixed points in self-similar analysis of time series”, Int. J.Mod. Phys. B , 1463–1476.Yukalov, V. I., Yukalova, E. P. & Sornette, D. [2012a] “Modeling symbiosis by interactions through speciescarrying capacities”, Physica D , 1270–1289.Yukalov, V. I., Yukalova, E. P. & Sornette, D. [2012b] “Extreme events in population dynamics withfunctional carrying capacity”, Eur. Phys. J. Spec. Top. , 313–354.Yukalov, V. I., Yukalova, E. P. & Sornette, D. [2013] “Utility rate equations of group population dynamicsin biological and social systems”, PLOS One , e83225.Yukalov, V. I., Yukalova, E. P. & Sornette, D. [2014] “Population dynamics with nonlinear delayed carryingcapacity”, Int. J. Bifur. Chaos , 1450021.ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES −1 −0.5 0 0.5 1 1.5 2 2.5−1−0.500.511.522.5 g b g c (b) (a) −5 −4 −3 −2 −1 0−18−15−12−9−6−30 bg −e b = gg (b)g (b) −e k = 1,2,3 { x k* , z k* } { x * , z * } Fig. 1. (a) The regions of existence of stationary states for passive symbiosis in the plane b − g . When b > < g < g c ( b ),there exist two fixed points, but only one is stable. If b ≤ g > 0, or b > g ≤ 0, there exists a single fixed pointthat is stable. If b, g < 0, there exist up to three fixed points, but not more then two of them are stable. (b) The region offixed-point existence under mutual parasitism, when b, g < 0. For either − e < b < g < 0, or for b < − e and g < g ( b ),or for b < − e and g ( b ) < g < 0, there exists only one fixed point that is stable. For b < − e and g ( b ) < g < g ( b ), there existthree fixed points, but only two of them are stable. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES t(a) z(t)x(t) t(b) x(t) z(t)0 10 20 30 40 50 60012345 tx(t) (c) z = 12.042205z = 12.042204 0 10 20 30 40 50 60024681012 tz(t) z = 12.042205z = 12.043304 (d) Fig. 2. Dynamics of populations in the case of mutualistic passive symbiosis for different symbiotic parameters b > g > 0, and different initial conditions: (a) For b = 1, g = 0 . > g c ( b ) ≈ . x = 1, z = 0 . 2, populations x ( t )(solid line) and z ( t ) (dashed line) monotonically increase, as t → ∞ ; (b) For the same b = 1, as in Fig. 2a, with the initialconditions x = 6 and z = 0 . 25, for different g taken slightly below and above the critical line g = g c ( b ). Below the criticalline, when g = 0 . < g c = 0 . x ( t ) tends to its stable fixed point x ∗ = 5 . 672 and the population z ( t ) tends monotonically from below to z ∗ = 1 . g = 0 . > g c , both populations, shown by dashed lines, tend to infinity, as t → ∞ ; (c) Dynamics of the population x ( t )for the symbiotic parameters b = 0 . 15 and g = 0 . < g c = 0 . x = 2, but for different initialconditions z . The population x ( t ) tends non-monotonically to its stable fixed point x ∗ = 1 . z is taken from thebasin of attraction of this point: x ( t ) → x ∗ for z = 12 (solid line), z = 12 . 04 (dashed line), z = 12 . x ( t ) → ∞ , as t → ∞ , when z = 12 . z ( t ) for the sameparameters b, g and the same initial conditions, as in Fig. 2c. The population converges to a stationary state, when the initialconditions are inside the attraction basin, and diverges, if the initial conditions are outside of this basin. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) g = 10g = 100 g = 1 0 1 2 3 4 5 60123456 tz(t) (b) g = 100 g = 10g = 1 Fig. 3. Dynamics of populations under passive symbiosis with one parasitic species. Influence of the symbiotic parameter g > x ( t ) , z ( t ), for the fixed b = − 1, and the initial conditions x = 3 and z = 1. Differentlines correspond to the varying parameter g : For g = 1 (dashed-dotted line) the stable fixed point is { x ∗ = 0 . , z ∗ = 1 . } ;for g = 10 (dashed line), the fixed point is { x ∗ = 0 . , z ∗ = 2 . } ; and for g = 100 (solid line), the stationary point is { x ∗ = 0 . , z ∗ = 4 . } . (a) Population x ( t ) tends monotonically from above to the corresponding stable stationary state, as t → ∞ ; (b) Population z ( t ) tends non-monotonically from below to the corresponding stable stationary state, as t → ∞ . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES −1 0 1 2 3 4 5−10123 tx(t) (a) tz(t) (b) Fig. 4. Dynamics of populations under passive symbiosis with one parasitic species. Influence of the initial condition z on thebehavior of solutions for the fixed b = − . g = 100, and x = 3. The corresponding stable fixed point is { x ∗ = 0 . , z ∗ =33 . } . (a) Population x ( t ) tends monotonically, as t → ∞ , from above to the stable stationary state x ∗ for z = 50 (solid line)and for z = 0 . z ( t ) tends, as t → ∞ , non-monotonically from above to the stable stationarystate z ∗ for z = 50 (solid line) and non-monotonically from below to z ∗ for z = 0 . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) g = − 0.25 g = − 1g = − 2.5 tz(t) g = − 0.25 g = − 1g = − 2.5 (b) tx(t) (c) b = − 10 b = − 2.5b = − 0.25 0 2 4 6 8 1000.20.40.60.811.2 tz(t) (d) b = − 0.25 b = − 2.5b = −10 Fig. 5. Dynamics of populations under passive symbiosis with two parasitic species. Influence of the symbiotic parameters b < g < { x = 2 , z = 1 } . (a) Dynamics of x ( t ) for b = − . g . The population x ( t ) monotonically tends to its stationary state: { x ∗ = 0 . , z ∗ = 0 . } for g = − . { x ∗ = 0 . , z ∗ = 0 . } for g = − { x ∗ = 0 . , z ∗ = 0 . } for g = − . z ( t ) for the same, as above, parameters b = − . g . Population z ( t ) monotonically, ornon-monotonically, from above tends to the corresponding stationary states. (c) Dynamics of x ( t ) for g = − b . The population x ( t ) monotonically tends from above to its stationary state: { x ∗ = 0 . , z ∗ = 0 . } for b = − . 25 (solid line); { x ∗ = 0 . , z ∗ = 0 . } for b = − . { x ∗ = 0 . × − , z ∗ = 0 . } for b = − z ( t ) for the same, as in Fig. 5c, parameters g = − b . Population z ( t )converges to the corresponding stationary states, either monotonically or non-monotonically. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) z = 0.1z = 5 z = 10 0 2 4 6 8 1000.511.52 tz(t) (b) z = 0.1 z = 5z = 100 2 4 6 8 1000.20.40.60.81 tx(t) (c) z = 0.15z = 0.5 z = 1.5 z = 10 0 2 4 6 8 1000.511.52 tz(t) (d) z = 0.1z = 0.5 z = 1.5z = 10 Fig. 6. Dynamics of populations under passive symbiosis with two parasitic species. Influence of initial conditions { x , z } on the behaviour of the parasitic populations with b = − g = − 2. The corresponding stable fixed point is { x ∗ =0 . , z ∗ = 0 . } . (a) Dynamics of x ( t ) for x = 1 . > x ∗ and different z . Population x ( t ) converges to the stationarystate x ∗ : monotonically for z = 0 . < z ∗ (solid line); non-monotonically for z = 5 (dashed line); and non-monotonically for z = 10 > z ∗ (dashed-dotted line); (b) Dynamics of z ( t ) for the same, as in Fig. 6a, initial condition x = 1 . z . Population z ( t ) tends to the stationary state z ∗ : non-monotonically for z = 0 . z = 5(dashed line); and monotonically for z = 10 (dashed-dotted line). (c) Dynamics of x ( t ) for x = 0 . < x ∗ and different z .Population x ( t ) converges to the stationary state x ∗ : monotonically for z = 0 . 15 (solid line); for z = 0 . z = 1 . z = 10 (dotted line). (d) Dynamics of z ( t ) for the same, as in Fig.6c, initial condition x = 0 . z . Population z ( t ) tends to the stationary state z ∗ : non-monotonically for z = 0 . z = 0 . z = 1 . z = 10 (dotted line). ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES b and g in qualitatively different regions: (a)Phase portrait for b = 0 . g = 0 . 3, where g > g c ( b ) ≃ . b = 0 . < g = 0 . < g c ( b ) ≃ . b, g , where two fixed points exist. The first fixed point { x ∗ = 2 . , z ∗ = 1 . } is a stable node, with theLyapunov exponents λ = − . λ = − . { x ∗ = 6 . , z ∗ = 3 . } is a saddle, with theLyapunov exponents λ = − . λ = 0 . b = 0 . g = − . 2. The parameters are in theregion of b, g , where only one fixed point exists, which is a stable focus { x ∗ = 1 . , z ∗ = 0 . } , with the characteristicexponents λ = − − . i , λ = − . i . (d) Phase portrait for b = g = − 3. The parameters are in the bistabilityregion of b, g , where three fixed points exist, two of them being stable and one, unstable. Here, the first stable fixed point isthe node { x ∗ = 0 . , z ∗ = 0 . } , with the Lyapunov exponents λ = − . λ = − . { x ∗ = z ∗ , z ∗ = x ∗ } , with the same Lyapunov exponents. And the third fixed point { x ∗ = z ∗ = 0 . } is the saddle, with the Lyapunov exponents λ = − . λ = 0 . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES g g = − b g = − b + 1/e1/e b Fig. 8. The regions of existence of the fixed-point for Eqs. (16) in the plane b − g . When b + g > /e , then there are no fixedpoints. If 0 < b + g < /e , then there exist two fixed points, one of them being stable and another, unstable. If b + g ≤ 0, thenthere exists a single fixed point that is stable. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES t ln x(t)ln z(t) (a) t ln x(t)ln z(t) (b) tx(t) (c) tz(t) (d) Fig. 9. Dynamics of populations under b + g > /e , with the initial conditions x = 0 . 01 and z = 5 for different symbioticparameters: (a) Logarithmic behavior of mutualistic populations for b = 2 and g = 1. Both populations grow, as t → ∞ , withln x ( t ) → ∞ increasing monotonically, while ln z ( t ) → ∞ , non-monotonically. (b) Logarithmic behavior of the populations,when one of them is parasitic, for b = − . g = 1. At t → ∞ , the host population dies out, ln x ( t ) → −∞ , that is, x ( t ) → z ( t ) → ∞ . (c) Dynamics of the host population x ( t ), for the sameparameters b = − . g = 1, demonstrating that x ( t ) → t → ∞ , in a non-monotonic way. (d) Dynamics ofthe parasitic population z ( t ), for the same parameters b = − . g = 1, showing the details of the non-monotonic increaseof z ( t ) → ∞ , as t → ∞ . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES t(a) z(t) x(t) t z(t) x(t) (b) t z(t) x(t) (c) Fig. 10. Population dynamics for 0 < b + g < /e , with the initial conditions x = 0 . 01 and z = 5. (a) For b = 0 . g = 0 . 1, the population x ( t ) → x ∗ = 1 . 668 monotonically from below, and population z ( t ) → z ∗ = z ∗ = 1 . 227 non-monotonically from above. (b) For b = − . 75 and g = 1, the population x ( t ) → x ∗ = 0 . z ( t ) → z ∗ = 4 . b = − g = − 5, the population x ( t ) → x ∗ = 0 . z ( t ) → z ∗ = 0 . 303 monotonically from above. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES b and g from qualitatively different regions: (a)Phase portrait for b = 0 . g = 0 . 3, where b + g > /e . The parameters are in the region, where there are no fixed points.(b) Phase portrait for b = 0 . g = 0 . 1, with 0 < b + g c ( b ) < /e . The parameters are in the region of b, g , where two fixedpoints exist. The first fixed point { x ∗ = 1 . , z ∗ = 1 . } is the stable node, with the Lyapunov exponents λ = − λ = − . { x ∗ = 3 . , z ∗ = 1 . } is a saddle, with the Lyapunov exponents λ = − λ = 0 . b = − . g = − 1. The parameters are in the region of b, g , where only one fixedpoint exists, which is the stable node { x ∗ = 0 . , z ∗ = 0 . } , with the Lyapunov exponents λ = − λ = − . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES −1 −0.5 0 0.5 1 1.5 2−0.3−0.2−0.100.10.20.30.40.5 bg g c (b)1/e b ≈ (b) Fig. 12. Regions of existence of the fixed-point for mixed symbiosis. For either 0 < b < /e and g > g c ( b ), or b > /e and g ≥ 0, Eqs. (21) do not have solutions, hence, there are no fixed points. For 0 < b < /e and 0 < g < g c ( b ), Eqs. (21) have twosolutions, but only one fixed point is stable. For b > /e and g c ( b ) < g < 0, there exists one unstable fixed point. When either0 < b < /e and g ( b ) < g < 0, or 1 /e < b < b ≈ . 47 and g ( b ) < g < g c ( b ), then there are three fixed points, but only oneof them is stable. When either 0 < b < b and g < g ( b ), or b ≥ b and g < g c ( b ), then there exists a single fixed point that isstable. When b ≤ g ∈ ( −∞ , ∞ ), then the unique solution to Eqs. (21) is a stable fixed point. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tln x (a) tln z (b) tx(t) (c) tz(t) (d) Fig. 13. Dynamics of populations under mixed symbiosis for the symbiotic parameters, when unbounded growth changes toeverlasting oscillations. The symbiotic parameter of the active species is fixed, b = 0 . > /e , while that of the passive speciesvaries. For the given b , the change of the behavior occurs on the line g = 0 that is higher than the Hopf bifurcation point g c ( b ) = − . { x = 5 , z = 0 . } . (a) Logarithmic behaviour of the active population x ( t ) for g = 0 . 05, where there are no fixed points. (b) Logarithmic behaviour of the passive population z ( t ) for g = 0 . 05. (c) Activespecies population x ( t ) oscillates without convergence for g = − . > g c ( b ) = − . z ( t )oscillates without convergence for the same g = − . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) tz(t) (b) tx(t) (c) tz(t) (d) Fig. 14. Dynamics of populations under mixed symbiosis for the symbiotic parameters, when everlasting oscillations changeto the oscillating convergence to a a stable focus. The symbiotic parameter of the active species is fixed, b = 0 . > /e , withthe Hopf bifurcation point g c ( b ) = − . { x = 5 , z = 0 . } . (a) Activespecies population x ( t ) oscillates with a small amplitude, as t → ∞ , when g = − . > g c ( b ) is slightly larger than the Hopfbifurcation point. The characteristic exponents for the unstable focus are λ = 0 . . i and λ = λ ∗ . (b) Passive speciespopulation z ( t ) oscillates, as t → ∞ , when g = − . > g c ( b ) is slightly larger than the Hopf bifurcation point. (c) Activespecies population x ( t ) converges with oscillations to its stable stationary state x ∗ = 7 . g = − . < g c ( b ) is slightlylower than the Hopf bifurcation point. The characteristic exponents for the stable focus are λ = − . . i and λ = λ ∗ .(d) Passive species population z ( t ) converges oscillating to its stable stationary state z ∗ = 0 . g = − . < g c ( b )is slightly lower than the Hopf bifurcation point. ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) g = 0.2g = − 2.5g = − 0.2 g = 0.164 0 5 10 15 20 2500.511.522.5 tz(t) (b) g = 0.2g = 0.164 g = − 0.2g = − 2.50 5 10 15 20 25 30048121620 tx(t) (c) g = − 0.2g = − 0.35g = − 1 g = − 2.5 0 5 10 15 20 25 3000.10.20.30.40.5 tz(t) (d) g = − 1 g = − 0.2g = − 0.35 g = − 2.5 Fig. 15. Population dynamics under mixed symbiosis for varying symbiotic parameters, with the initial conditions { x =5 , z = 0 . } . (a) Active species population dynamics for the fixed b = 0 . < /e , when g c ( b ) = 0 . g ( b ) = − . g = − . < g ( b ) (solid line), when the population x ( t ) tends to x ∗ = 1 . g = − . x ( t ) → x ∗ = 1 . 28; 0 < g = 0 . < g c ( b ) (dotted line), when the population x ( t ) → x ∗ = 2 . 03; and g = 0 . > g c ( b ) (dashed-dotted line), when x ( t ) → ∞ , as t → ∞ . (b) Passive species populationdynamics for b = 0 . 25 and the same initial conditions, with the varying symbiotic parameter g of the passive species: g = − . z ( t ) → z ∗ = 0 . g = − . z ( t ) → z ∗ = 0 . g = 0 . 164 (dotted line), when z ( t ) → z ∗ = 1 . 39; and g = 0 . > g c ( b ) (dashed-dotted line), when x ( t ) → ∞ , as t → ∞ . (c) Active species population dynamicsunder the fixed b = 2 > b ≈ . 47, with g c ( b ) = − . g : g = − . x ( t ) → x ∗ = 1 . g = − x ( t ) → x ∗ = 1 . g = − . < g c ( b ) (solid line), when x ( t ) → x ∗ = 5 . 28 with a fewoscillations; and g = − . > g c ( b ) (dashed line), when x ( t ) oscillates without convergence, as t → ∞ . (d) Passive speciespopulation dynamics for b = 2 and the same initial conditions, with the varying symbiotic parameter g : g = − . z ( t ) → z ∗ = 0 . g = − z ( t ) → z ∗ = 0 . g = − . < g c ( b ) (solid line), when z ( t ) → z ∗ = 0 . 157 with a few oscillations, and g = − . > g c ( b ) (dashed line), with z ( t ) oscillating without convergence, as t → ∞ . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) (a) g = − 3 g = 5 g = 2g = − 1 0 3 6 9 12 150246810 tz(t) (b) g = 5 g = 2 g = −1g = − 30 5 10 15 20 25 3002468101214 tx(t) (c) b = 20b = 2 b = 0 b = − 2 0 5 10 15 20 25 3000.30.60.91.21.5 tz(t) (d) b = − 2 b = 0b = 2b = 20 Fig. 16. Population dynamics under mixed symbiosis for varying symbiotic parameters, with the initial conditions { x =0 . , z = 5 } . (a) Active species population dynamics under the fixed b = − . 25 and different g : g = 5 (solid line), when x ( t ) → x ∗ = 0 . g = 2 (dashed line), with x ( t ) → x ∗ = 0 . g = − x ( t ) → x ∗ = 0 . g = − x ( t ) → x ∗ = 0 . b = − . 25 and different g : g = 5 (solid line), with z ( t ) → z ∗ = 8 . 21 non-monotonically from below; g = 2 (dashed line), when z ( t ) → z ∗ = 3 . g = − z ( t ) → z ∗ = 0 . g = − z ( t ) → z ∗ = 0 . g = − . b : b = − x ( t ) → x ∗ = 0 . b = 0 (dotted line), with x ( t ) → x ∗ = 1 monotonically from below; b = 2 (dashed line), when g c ( b ) = − . > g , and x ( t ) → x ∗ = 3 . 43 with a few oscillations; b = 20 (dashed-dotted line),when g c ( b ) = − . < g < 0, and solution x ( t ) oscillates without convergence, as t → ∞ . (d) Passive species populationdynamics under the fixed g = − . b : b = − z ( t ) → z ∗ = 0 . b = 0 (dotted line), with z ( t ) → z ∗ = 0 . 607 monotonically from above; b = 2 (dashed line), when z ( t ) → z ∗ = 0 . b = 20(dashed-dotted line), when z ( t ) oscillates without convergence, as t → ∞ . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES t ln z(t)ln x(t) (a) t ln z(t)ln x(t) (b) Fig. 17. Logarithmic behaviour of populations x ( t ) (solid line) and z ( t ) (dashed-dotted line) for b = 0 . 4, and the initialconditions { x = 5 , z = 0 . } , when g c ( b ) ≈ − . g ( b ) ≈ − . g c ( b ) < g = − . < 0. (b) Everlasting oscillations of populations for g ( b ) < g = − . < g c ( b ). ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES Fig. 18. Phase portrait for mixed symbiosis, with the symbiotic parameters b and g from qualitatively different regions: (a)Phase portrait for the symbiotic parameters, where there are no fixed points. The parameters are b = 0 . g = 0 . > g c ( b ).Here g c ( b ) ≈ . b = 0 . g = 0 . < g c ( b ), with g c ( b ) ≈ . { x ∗ = 1 . , z ∗ = 1 . } is stable, with the Lyapunov exponents λ = − . λ = − . { x ∗ = 2 . , z ∗ = 3 . } is a saddle, with the Lyapunov exponents λ = − . λ = 0 . b = − . g = 1, where there is a single fixedpoint { x ∗ = 0 . , z ∗ = 1 . } that is a stable focus, with the characteristic exponents λ = − . − . i and λ = λ ∗ ). ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES b > g < b = 0 . g = − . < g c ( b ) < 0, with g c ( b ) ≈ − . { x ∗ = 3 . , z ∗ = 0 . } that is a stable focus, with the characteristic exponents λ = − . − . i and λ = λ ∗ . (b) Phase portrait for the same b = 0 . 5, but with g c ( b ) < g = − . < 0. In this region, there exist a singlefixed point { x ∗ = 8 . , z ∗ = 0 . } , which is an unstable focus, with the characteristic exponents λ = 0 . − . i , λ = λ ∗ , and a limit cycle. (c) Phase portrait for b = 0 . g = − . g c ( b ) ≈ − . g ( b ) ≈ − . g ( b ) < g < g c ( b ) < 0. The parameters are in the region of b > , g < 0, where three fixed points exist. One ofthem, { x ∗ = 2 . , z ∗ = 0 . } is a stable node, with the Lyapunov exponents λ = − . λ = − . { x ∗ = 8 . , z ∗ = 0 . } is a saddle, with the Lyapunov exponents λ = − . λ = 0 . { x ∗ = 13 . , z ∗ = 0 . } is an unstable focus, with the characteristic exponents λ = 0 . − . i and λ = λ ∗ . ugust 4, 2014 0:17 symbiosis˙Subm˙ARX REFERENCES tx(t) b = 0.5 b = 0.3b = −1 b = 0 (a) tz(t) (b) g = 0g = −0.5g = 0.5 Fig. 20. Dynamics of populations under commensalism, characterized by the degenerate systems of equations (24), (26), and(27). Initial conditions are { x = 5 , z = 0 . } . (a) Dynamics of the population x ( t ) for different b and g = 0: For b = 0 (dottedline), x ( t ) → x ∗ = 1; for b = 0 . > /e (dashed-dotted line), x ( t ) → ∞ ; for b = 0 . < /e (dashed line), x ( t ) → x ∗ = 1 . b = − x ( t ) → x ∗ = 0 . t → ∞ . (b) Dynamics of the population z ( t ) for different g and b = 0: For g = 0 (dotted line), z ( t ) → z ∗ = 1; for g = 0 . z ( t ) → z ∗ = e g = 1 . 65; and for g = − . z ( t ) → z ∗ = 0 . t → ∞→ ∞