Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation
aa r X i v : . [ q - b i o . P E ] A p r October 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
DYNAMIC TRANSITION IN SYMBIOTIC EVOLUTIONINDUCED BY GROWTH RATE VARIATION
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)
In a standard bifurcation of a dynamical system, the stationary points (or more generally attrac-tors) change qualitatively when varying a control parameter. Here we describe a novel unusualeffect, when the change of a parameter, e.g. a growth rate, does not influence the stationarystates, but nevertheless leads to a qualitative change of dynamics. For instance, such a dynamictransition can be between the convergence to a stationary state and a strong increase withoutstationary states, or between the convergence to one stationary state and that to a differentstate. This effect is illustrated for a dynamical system describing two symbiotic populations,one of which exhibits a growth rate larger than the other one. We show that, although the sta-tionary states of the dynamical system do not depend on the growth rates, the latter influencethe boundary of the basins of attraction. This change of the basins of attraction explains thisunusual effect of the quantitative change of dynamics by growth rate variation.
Keywords : Dynamics of symbiotic populations, growth rate, functional carrying capacity, dy-namic transitions, basin of attraction, bifurcation. ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette
1. Introduction
It is well known that varying the parameters controlling a dynamical system can change the existing fixedpoints. In the case of a bifurcation, this can qualitatively change the dynamical behavior of the system,leading to what is often called a dynamic phase transition or bifurcation transition [Schuster, 1984]. Whenthe considered parameter characterizes a growth rate, its variance usually leads just to the acceleration orslowing down of the convergence towards the stable fixed points, but does not induce dynamic transitions.In the present paper, we show that this common wisdom is not always correct. It may happen that avarying growth rate, while not influencing the fixed points, can nevertheless induce qualitative changesin the dynamics similar to a bifurcation transition, while no bifurcation of stationary states occurs. Wedemonstrate this unusual effect by considering an autonomous dynamical system describing co-evolvingsymbiotic populations.Qualitatively, the fact that the evolution of symbiotic species essentially depends on their proliferationrates has been discussed in many publications. For example, it is known that, for optimal development,mutualistic symbiotic species “must keep pace” between each other [Bennett & Moran, 2015]. The growthrate of fungal endophites can either enhance or reduce plant reproduction [Rodriguez et al. , 2009]. Reefcorals engage in symbiosis with single-celled Dinoflagelate Algae, from which they acquire photosyntheticproducts that support most of their energetic needs and help them build calcium carbonate skeletons thatform the foundation of coral reefs. Unsufficient growth of the Algae results in the increased coral bleachingand mortality [Cunning & Baker, 2014]. Intense proliferation of viral pathogens, such as the DeformedWing Virus, undermines honey bee colonies and can lead to their collapse [Di Prisco et al. , 2016]. Thesymbiosis that is the most important for humans is the one between the human body and the multitudesof about 10 microorganisms, consisting of bacteria, archaea, and fungi, participating in the synthesis ofessential vitamins and amino acids, as well as in the degradation of otherwise indigestible plant materialand of certain drugs and pollutants in the guts [Ley et al. , 2006]. It is now known that our gut microbiomecoevolves with us and that their evolution can have major consequences, both beneficial and harmful,for human health [Ley et al. , 2008]. It is well established that mycorrhizal fungi symbiosis with plants isbeneficial for plant growth and reproduction. However too fast proliferation of the fungi at the early stageof the plant seedling can have negative effects because of the carbon costs associated with sustaining thefungi [Varga & Kyt¨oviita, 2016].Usually, in a symbiotic coexistence, the faster growth of species has just the effect of a faster convergenceto the stationary states. Although in some cases, the change of a growth rate can result in a differentstate. We suggest a mathematical model demonstrating the existence of the unusual effect of a qualitativechange of dynamical behavior induced by the variation of growth rates, while the stationary points are leftuntouched. Strictly speaking, this effect can occur in different nonlinear dynamical systems with feedbacks.We suggest a symbiotic interpretation for concretness and for explaining that the effect can really occurin nature. Section 2 presents the model. Section 3 studies the stable stationary states. Section 4 reviewsthe cases where a change of a growth rate only modifies the rate of convergence to the stationary states.Section 5 covers the cases where the change of a growth rate leads to dynamic transitions. Section 6describes the scale-separation approach that provides approximate solutions of the equations in the limitof large differences between the growth rates of the two species. Section 7 summarizes the article andconcludes by suggesting a biological fungi-plant system in which the reported effect could be at work.
2. Symbiosis with Functional Carrying Capacity
Symbiotic species interact with each other through influencing their carrying capacities [Boucher, 1988;Douglas, 1984; Sapp, 1994; Ahmadjian & Paracer, 2000]. A mathematical model characterizing these in-teractions has been suggested in [Yukalov et al. , 2012a,b, 2014a,b, 2015], where a detailed justification anddiscussions on numerous possible applications for biological and social symbiotic systems can be found. Inthese previous articles, symbiotic species were assumed to enjoy the same growth rate. Here, we analyzethe influence of the birth rates on the behavior of the populations. It turns out that changing birth ratesnot merely modifies the velocity of the growth processes, but can also lead to the unexpected effect of adrastic change in the dynamics of populations.ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December
Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation Let us consider symbiotic species, enumerated by the index i and whose populations are denoted by N i . Each population satisfies the logistic-type equation dN i dt = γ i (cid:18) N i − N i K i (cid:19) , (1)where γ i is a birth rate and K i is the carrying capacity, generally being a functional of the populations[Yukalov et al. , 2012a,b, 2014a]. By employing a scaling parameter C i , it is always possible to introducedimensionless quantities for each of the populations and for the related carrying capacity, respectively, x i ≡ N i C i , y i ≡ K i C i . (2)Then equation (1) reads as dx i dt = γ i (cid:18) x i − x i y i (cid:19) . (3)The explicit expression for the carrying capacity can be derived in the following way. Keeping in mindthat the carrying capacity y i is a function of the dimensionless populations x i , it is possible to express itas a Taylor expansion y i = y i ( x , x , . . . ) , y i ≃ X j c j x j + X kl c kl x k x l . Note that the first term of the expansion can be made equal to 1 by the appropriate choice of the scalingparameter C i . When the values of x i are small, it is admissible to limit oneself to a finite number of terms inthe above expansion. However, the assumption of the smallness of x i is too restrictive. The generalizationto arbitrary values of the variables x i can be accomplished by resorting to the self-similar approximationtheory [Yukalov, 1991, 1992], providing an effective summation of the infinite series. Using exponentialself-similar summation [Yukalov & Gluzman, 1998] we obtain y i = exp X j a j x j . (4)The growth rate γ i can be presented as the difference γ i = γ birth − γ death of a birth rate and a deathrate. In what follows, we assume that the birth rate surpasses the death rate, so that the growth rate ispositive, γ i > α ≡ γ γ . (5)To simplify the notation, we denote x ≡ x , z ≡ x (6)and measure time in units of 1 /γ . Thus we come to the two-dymensional dynamical system describing thesymbiosis of the dimensionless populations x and z , with the equations dxdt = α (cid:18) x − x y (cid:19) (7)and dzdt = z − z y . (8)The mutual carrying capacities, in the case of symbiosis, depend on the populations of the other species.The species self-action is excluded, since it is related to other effects influencing the carrying capacity byctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette self-improvement or self- destruction, which are not connected to symbiosis [Yukalov et al. , 2009, 2012b].We set the notation a = b and a = g . Then the carrying capacities take the form y = e bz , y = e gx . (9)Since our aim is to analyze the dynamics under different growth rates, we can assume, without loss ofgenerality, that γ is larger than γ , so that α > γ > γ ) . (10)In particular, α can be much larger than one, which would classify the variable x as fast and z as slow.It is useful to emphasize that the system of equations (7) and (8) describes all types of symbiosis,depending on the symbiotic parameters b and g . Thus, mutualism corresponds to the case b > , g > mutualism ) . Parasitic symbiosis is characterized by one of the inequalities b > , g < b < , g > b < , g < ( parasitism ) . And commensalism happens under one of the conditions b > , g = 0 b = 0 , g > (cid:27) ( commensalism ) . This classification derives from the fact that the signs of the parameters b and g define whether the mutualinfluence on the carrying capacities is beneficial (positive sign) or destructive (negative sign). While a zeroparameter signifies the absence of influence.
3. Evolutionary Stable Stationary States
The dynamical system under consideration is given by the equations dxdt = α (cid:16) x − x e − bz (cid:17) , dzdt = z − z e − gx , (11)with the parameters spanning the following intervals − ∞ < b < ∞ , −∞ < g < ∞ , α > . (12)We are looking for non-negative solutions x = x ( t ) ≥ z = z ( t ) ≥
0, with initial conditions x = x (0) , z = z (0) . There are three trivial fixed points: the unstable node { , } , with the characteristic exponents λ = 1and λ = α ; a saddle { , } , with the characteristic exponents λ = 1 and λ = − α ; and the saddle { , } ,with the characteristic exponents λ = − λ = α .The nontrivial stationary states are defined by the equations x ∗ = e bz ∗ , z ∗ = e gx ∗ , (13)which can also be represented as x ∗ = exp (cid:16) be gx ∗ (cid:17) , z ∗ = exp (cid:16) ge bz ∗ (cid:17) . It is important to stress that the stationary states, defined by equations (13), do not depend on thegrowth rate α .The characteristic exponents are the solutions to the equation λ + c λ + c = 0 , ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation where c = λ λ = α (1 − bgx ∗ z ∗ ) , c = − ( λ + λ ) = 1 + α . Thus λ , = −
12 (1 + α ) ± p (1 − α ) + 4 αbgx ∗ z ∗ . (14)The plane of the parameters b and g is separated into five regions with different behavior of thesolutions.In the region of strong mutualism A = { b > , g > g c ( b ) } , (15)there are no fixed points.In the region of moderate mutualism B = { b > , < g < g c ( b ) } , (16)there are two fixed points, a stable node { x ∗ , z ∗ } , with a limited basin of attraction, and a saddle { x ∗ , z ∗ } ,such that 1 < x ∗ < x ∗ , < z ∗ < z ∗ . The region of one-side parasitism C = (cid:26) b < , g > b > , g < , (17)where one of the species is parasitic, while the other is not, contains a stable focus { x ∗ , z ∗ } , with the basinof attraction being the whole plane of initial conditions x and z .The region of two-side parasitism D = { b < , g < } = D [ D , (18)where both species are parasitic, is divided into two subregions. In the subregion D = b < − e, g < g ( b ) < − eb < − e, g ( b ) < g < − e < b < , g < , (19)there exists only one stable node, with the basin of attraction being the whole plane of initial conditions x and z . While the subregion D = { b < − e , g ( b ) < g < g ( b ) < − e } = D \ D (20)contains a stable node { x ∗ , z ∗ } , with a limited basin of attraction, a saddle { x ∗ , z ∗ } , and another stablenode { x ∗ , z ∗ } , with a limited basin of attraction. The fixed points are related by the inequalities1 > x ∗ > x ∗ > x ∗ , z ∗ < z ∗ < z ∗ < . These regions are shown in Fig. 1.
4. Growth-Rate Acceleration of Population Dynamics
Since the stationary states, defined by equations (13), do not depend on the growth rate α , it is reasonableto expect that the increase of the latter should result only in the acceleration of the temporal dynamicsof the symbiotic populations, without qualitative changes in the overall picture. In many cases, it is reallyso, as is explained below. Strong mutualistic growth of populations
In region A , where there are no fixed points, mutualistic populations grow faster when increasing thegrowth rate α , displaying the same qualitative behavior, as is illustrated in Fig. 2.ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette −1 −0.5 0 0.5 1 1.5 2 2.5−1−0.500.511.522.5 b ABC CD +D g c (b) g −5 −4 −3 −2 −1 0−18−15−12−9−6−30 bg D D D b = gg (b) g (b)b = g −e−e Fig. 1. Regions on the plane b − g , as discussed in the text. In region A , there are no fixed points. In region B , there existtwo fixed points, one being a stable node, while the other is a saddle. Region C contains one fixed point being a stable focus.Region D is subdivided into two subregions shown in more details in the right panel. In region D , there is one stable fixedpoint, being a stable node. In region D , there are three fixed points, two of them being stable nodes, while the third is asaddle. tx(t) (a) α = 10 α = 1 α = 100 tz(t) α = 100 α = 1 (b) α = 10 Fig. 2. Dynamics of populations x ( t ) and z ( t ) in the parametric region A , for different growth rates. Here b = 1 and g = 0 . > g c ≈ . { x = 1 , z = 0 . } . (a) Population x ( t ) for α = 1 (solid line), α = 10(dashed-dotted line), and α = 100 (dashed line); (b) population z ( t ) for α = 1 (solid line), α = 10 (dashed-dotted line), and α = 100 (dashed line). Convergence to single stationary states
In region C , there is just a single fixed point, being a stable focus. The convergence to the stationary statecan be of slightly different type, as is shown in Figs. 3 and 4, but it is always faster when the parameter α is larger. The phase portrait is presented in Fig. 5.The region D contains a single stationary state, a stable node. Again, the convergence to the stationarystate is faster when the growth rate α is larger, as is shown in Fig. 6. Convergent behavior in marginal cases
The marginal cases correspond to zero values of symbiotic parameters. Thus, if b = 0, while g is arbitrary,the sole stationary state is the stable node x ∗ = 1 , z ∗ = e g ( b = 0 , − ∞ < g < ∞ ) , with the characteristic exponents λ = − λ = − α . The population x is described by the explicitformula x = x x + (1 − x ) exp( − αt ) . The convergence to the stationary state is faster for larger α .ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation tx(t) α = 10 α = 1 α = 100 (a) tz(t) (b) α = 1 α = 100 α = 10 tx(t) (c) α = 10 α = 1 α = 100 tz(t) (d) α = 1 α = 100 α = 10 Fig. 3. Dynamics of populations x ( t ) and z ( t ) in the parametric region C , for different b < g = 100 >
0. The initialconditions are x = 3 and z = 1. (a) Population x ( t ), with b = − .
1, for α = 1 (solid line), α = 10 (dashed line), and α = 100(dashed-dotted line). The stable fixed point is x ∗ = 0 . z ( t ), with b = − .
1, for α = 1 (solid line), α = 10(dashed line), and α = 100 (dashed-dotted line). The stable fixed point is z ∗ = 33 . x ( t ), with b = − α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). The stable fixed point is x ∗ = 0 . z ( t ), with b = −
1, for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). The stable fixedpoint is z ∗ = 4 . When b is arbitrary, while g = 0, then again there exits just a single fixed point, a stable node x ∗ = e b , z ∗ = 1 , with the characteristic exponents λ = − α and λ = −
1. The population z does not depend on α , beinggiven by the expression z = z z + (1 − z ) exp( − t ) , while the population x converges to the stationary state faster for larger α .The examples of the present section illustrate the expected situation, where the growth rate α directlyinfluences the time scales of the dynamics of the symbiotic populations, but does not qualitatively distortthe overall picture.
5. Growth-Rate Induced Dynamic Transitions
In the present section, we show that there may happen unexpected situations, when the variation of thegrowth rate, although not influencing the stationary states, can lead to dramatic changes in the populationdynamics.
Dynamic transition under mutualism
In the parametric region B , where b > < g < g c ( b ), there exist two fixed points, { x ∗ , z ∗ } and { x ∗ , z ∗ } , with 1 < x ∗ < x ∗ and 1 < z ∗ < z ∗ . The fixed point { x ∗ , z ∗ } is a stable node and { x ∗ , z ∗ } is asaddle. The stable node possesses a basin of attraction, whose boundary passes through the saddle. Thectober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette tx(t) (a) α = 1 α = 10 α = 100 tz(t) (b) α = 1 α = 10 α = 100 tx(t) (c) α = 100 α = 1 α = 10 tz(t) (d) α = 1 α = 10 α = 100 Fig. 4. Dynamics of populations x ( t ) and z ( t ) in the parametric region C for different initial conditions and different growthrates, when b >
0, while g <
0. (a) Population x ( t ), with the symbiotic parameters b = 100 and g = −
1. The initial conditionsare x = 1 and z = 0 .
02. The growth rate is α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). Thestable fixed point is x ∗ = 4 . z ( t ) for the same symbiotic parameters and initial conditions as in (a) for thegrowth rates α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). The stationary state is z ∗ = 0 . x ( t ), with the symbiotic parameters b = 1 and g = − .
5. The initial conditions are x = 3 and z = 1. Growthrate is α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). The stationary state is x ∗ = 1 . z ( t ) for the same symbiotic parameters and initial conditions, as in (c), for α = 1 (solid line), α = 10 (dashed line),and α = 100 (dashed-dotted line). The stationary state is z ∗ = 0 . xz (a) xz (b) Fig. 5. Phase portrait on the plane x − z , for the parametric region C , for b = − g = 1, and different growth rates α .There exists a single fixed point, a stable focus, shown by the filled green disc. The fixed point is { x ∗ = 0 . , z ∗ = 1 . } .(a) Phase portrait for α = 1; (b) phase portrait for α = 10. behavior of the populations x ( t ) and z ( t ) depends on whether the initial conditions are taken inside thebasin of attraction or not.On the line { b, g c ( b ) } , the stable node { x ∗ , z ∗ } , and the saddle { x ∗ , z ∗ } merge together and disappearfor g > g c ( b ). When g = g c ( b ), then x ∗ = x ∗ and z ∗ = z ∗ .It turns out that the growth rate α , although not influencing the stationary states as such, doesctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation tx(t) (a) α = 1 α = 100 α = 10 tz(t) (b) α = 1 α = 100 α = 10 tx(t) α = 10 α = 1 α = 100 (c) tz(t) (d) α = 1 α = 10 α = 100 Fig. 6. Dynamics of populations x ( t ) and z ( t ) in the region D , for different symbiotic parameters and initial conditions. (a)Population x ( t ), with b = − . g = −
1, and the initial conditions { x = 2 , z = 1 } , for α = 1 (solid line), α = 10 (dashedline), and α = 100 (dashed-dotted line). The stable fixed point is x ∗ = 0 . z ( t ), for the same symbioticparameters and initial conditions, as in (a), for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line).The fixed point is z ∗ = 0 . x ( t ), with b = − g = −
2, initial conditions { x = 0 . , z = 5 } , for α = 1(solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). The fixed point is x ∗ = 0 . z ( t )for the same symbiotic parameters and initial conditions, as in (c), for α = 1 (solid line), α = 10 (dashed line), and α = 100(dashed-dotted line). The fixed point is z ∗ = 0 . influence the boundary of the attraction basin. Therefore, it may happen that the same initial conditions,depending on the value of α , can occur inside the attraction basin or outside it. This delicate situation isillustrated in Figs. 7 to 9.Figure 7 demonstrates the convergence of the populations x ( t ) and z ( t ) for g taken close to the line { b, g c ( b ) } , with initial conditions that are inside the attraction basin of the stable fixed point for all α = 1 , , { x , z } are such that they are outside of the basin ofattraction for α = 1, but inside it, when α = 10 and α = 100. Contrary to Fig. 8, in Fig. 9, we show thesituation when the initial conditions { x , z } are inside the basin of attraction of the stable fixed pointfor α = 1, but outside of it for α = 10 and α = 100. Phase portraits for region B , under different α , arepresented in Fig. 10. The boundary of the attraction basin essentially depends on the symbiotic parameters b and g , as well as on the growth rate α . Dynamic transition under parasitism
In the parametric region D , there exist three fixed points, such that 1 > x ∗ > x ∗ > x ∗ and z ∗ < z ∗ 1. The points { x ∗ , z ∗ } and { x ∗ , z ∗ } are stable nodes, while { x ∗ , z ∗ } is a saddle. In the region D , thebehavior of populations depends on initial conditions { x , z } and on the growth rate α . With increasingtime, the populations x ( t ) and z ( t ) can tend either to { x ∗ , z ∗ } or to { x ∗ , z ∗ } , depending on the choseninitial conditions. The location of the boundary between the attraction basins, corresponding to differentfixed points, strongly depends on the growth rate α . The temporal behavior of the symbiotic populationsis illustrated in Figs. 11, 12 and 13. Figures 14 and 15 present the related phase portraits for differentgrowth rates.ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette tx(t) (a) α = 1 α = 10 α = 100 tz(t) α = 1 α = 10 α = 100 (b) Fig. 7. Dynamics of populations x ( t ) and z ( t ) in the parametric region B , with b = 1 and g = 0 . < g c ≈ . { x = 6 , z = 0 . } , such that the initial point is inside the attraction basin for all α = 1 , , { x ∗ = 5 . , z ∗ = 1 . } , and the saddle is { x ∗ = 5 . , z ∗ = 1 . } . (a) Population x ( t ) for α = 1 (solidline), α = 10 (dashed line), and α = 100 (dashed-dotted line); (b) population z ( t ) for α = 1 (solid line), α = 10 (dashed line),and α = 100 (dashed-dotted line). tx(t) α = 1 α = 100 (a) α = 10 tz(t) α = 100 α = 1 α = 10 (b) Fig. 8. Dynamics of populations x ( t ) and z ( t ) for the parametric region B , with b = 1 and g = 0 . < g c ≈ . { x ∗ = 5 . , z ∗ = 1 . } . The saddle is { x ∗ = 5 . , z ∗ = 1 . } , also as in figure 7.The initial conditions { x = 6 . , z = 1 . } are such that they are outside of the attraction basin for α = 1, but inside for α = 10 and α = 100. (a) Population x ( t ) for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line); (b)population z ( t ) for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). tx(t) (a) α = 1 z = 7.05090741z = 7.05090742 α = 10 tz(t) (b) α = 10 z = 7.05090741z = 7.05090742 α = 1 Fig. 9. Dynamics of populations x ( t ) and z ( t ) in the parametric region B , with b = 0 . 15 and g = 0 . < g c ≈ . x = 2 is fixed and z is varied close to the boundaries of the attraction basins. The stable fixed point is { x ∗ = 1 . , z ∗ = 2 . } , and the saddle is { x ∗ = 2 . , z ∗ = 6 . } . (a) Population x ( t ) for α = 1 and z = 7 . α = 10 and z = 7 . α = 10 but z = 7 . z ( t ) for α = 1 and z = 7 . α = 10 and z = 7 . α = 10 but z = 7 . ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation xz (a) xz (b) xz (c) Fig. 10. Phase portraits in the plane x − z for the parametric region B , for the symbiotic parameters b = 0 . 15 and g = 0 . α . The stable node { x ∗ = 1 . , z ∗ = 2 . } is denoted by the green filled disc, and the saddle { x ∗ = 2 . , z ∗ = 6 . } , by a red filled rectangle. The boundary of the basin of attraction is shown by the dashed line. (a)Phase portrait for α = 1; (b) phase portrait for α = 10; (c) phase portrait for α = 500. tx(t) x α = 100 α = 10 α = 1 (a) tz(t) z (b) α = 100 α = 10 α = 1 tx(t) (c) α = 1 α = 10 α = 100 x tz(t) z (d) α = 1 α = 10 α = 100 Fig. 11. Dynamics of populations x ( t ) and z ( t ), in the parametric region D , for the symbiotic parameters b = − . g = − 4, with the initial condition x = 0 . 171 and different z and α . There are two stable nodes, { x ∗ = 0 . , z ∗ = 0 . } and { x ∗ = 0 . , z ∗ = 0 . } , and the saddle { x ∗ = 0 . , z ∗ = 0 . } . (a) Population x ( t ), with z = 0 . > z ∗ ,for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line); (b) population z ( t ), with the same z , as in(a), for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line); (c) population x ( t ), with z = 0 . < z ∗ ,for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line); (d) population z ( t ), with the same z , as in(c), for α = 1 (solid line), α = 10 (dashed line), and α = 100 (dashed-dotted line). 6. Approximate Solutions of Symbiotic Equations in the Presence of CoexistingFast and Slow Populations For large growth rates α ≫ 1, equations (11) imply that the variable x is fast while the variable z is slow.In this case, the analysis of the evolution equations can be done by resorting to the Bogolubov-Krylovaveraging techniques [Bogolubov & Mitropolsky, 1961]. As is described in the scale-separation approach[Yukalov, 1993], we solve the equation for the fast variable, keeping the slow variable as a quasi-integral ofctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette tx(t) x α = 10 α = 1 (a) x tz(t) z z (b) α = 10 α = 1 tx(t) x (c) α = 1 α = 1.35 α = 1.36 α = 10 x tz(t) (d) z α = 1 α = 10 α = 1.35 α = 1.36 z Fig. 12. Dynamics of populations x ( t ) and z ( t ), in the parametric region D , with b = − . g = − x = 0 . 25 and z = 0 . α . The stable nodes are { x ∗ = 0 . , z ∗ = 0 . } and { x ∗ = 0 . , z ∗ = 0 . } and the saddle is { x ∗ = 0 . , z ∗ = 0 . } , as in figure 11. (a) Population x ( t ) for α = 1(solid line) and α = 10 (dashed line); (b) population z ( t ) for α = 1 (solid line) and α = 10 (dashed line); (c) population x ( t )for α = 1 (thin solid line), α = 1 . 35 (solid line), α = 1 . 36 (dashed-dotted line), and α = 10 (thin dashed line); (d) population z ( t ) for α = 1 (thin solid line), α = 1 . 35 (solid line), α = 1 . 36 (dashed-dotted line), and α = 10 (thin dashed line). For thegiven parameters, there exists the critical value α crit of the growth rate, such that 1 . < α crit < . 36. Under the same initialconditions, if 1 ≤ α < α crit , then the populations tend to the node { x ∗ = 0 . , z ∗ = 0 . } , while when α > α crit , thepopulations converge to the other node { x ∗ = 0 . , z ∗ = 0 . } . motion, which gives x = x x (1 − e − αt ) e − bz + e − αt . (21)This expression is substituted into the equation for the slow variable, which is averaged over time, resultingin the equation dzdt = z − z exp (cid:16) − ge bz (cid:17) . (22)Equations (21) and (21) define the so-called guiding centers of the solutions to equations (11).In Figures 16 and 17, we demonstrate that the guiding-center solutions, prescribed by equations (21)and (22), provide rather good approximations for the exact solutions following from the initial equations(11). Surprisingly, the approximate solutions are already rather close to the true solutions even for α = 1.The stationary states are identical for the approximate solutions and for exact ones. 7. Conclusion In the case of a standard dynamic transition, a qualitative change of dynamical behavior occurs when asystem parameter reaches a bifurcation point, where the nature of fixed points changes. We have demon-strated the existence of a non-standard dynamic transition, in which a qualitative change of dynamicalbehavior occurs as a result of the variation of the growth rate that does not influence the fixed points.The sharp change in dynamical behavior happens because the varying growth rate shifts the boundary ofthe basins of attraction of the fixed points, while the fixed points themselves do not change. Typically, thectober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation tx(t) x x α = 10 α = 1 α = 100 (a) tz(t) (b) α = 1 α = 10 α = 100 z z tx(t) (c) x α = 37.06 α = 100 α = 1 α = 37.05 x tz(t) α = 100 α = 1 α = 37.05 α = 37.06z z (d) Fig. 13. Dynamics of populations x ( t ) and z ( t ), in the parametric region D , with b = − . g = − 4, the initial conditions x = 0 . z = 0 . 5, for different α . The stable nodes are { x ∗ = 0 . , z ∗ = 0 . } and { x ∗ = 0 . , z ∗ = 0 . } and the saddle is { x ∗ = 0 . , z ∗ = 0 . } , as in figs. 11 and 12. (a) Population x ( t ) for α = 1 (solid line), α = 10(dashed line), and α = 100 (dashed-dotted line); (b) population z ( t ) for α = 1 (solid line), α = 10 (dashed line), and α = 100(dashed-dotted line); (c) population x ( t ) for α = 1 (thin solid line), α = 37 . 05 (solid line), α = 37 . 06 (dashed-dotted line), and α = 100 (thin dashed line); (d) population z ( t ) for α = 1 (thin solid line), α = 37 . 05 (solid line), α = 37 . 06 (dashed-dottedline), and α = 100 (thin dashed line). There exists the critical growth rate α crit in the interval 37 . < α crit < . 06, suchthat, when 1 ≤ α < α crit , the populations tend to the node { x ∗ = 0 . , z ∗ = 0 . } , while if α > α crit , the populationsconverge to the node { x ∗ = 0 . , z ∗ = 0 . } . xz (a) xz (b) xz (c) Fig. 14. Phase portraits on the plane x − z for the parametric region D , with b = g = − 3, for different α . The dashed lineshows the boundary between the attraction basins of the stable nodes { x ∗ = 0 . , z ∗ = 0 . } and { x ∗ = 0 . , z ∗ =0 . } , which are represented by the filled green discs. The saddle point { x ∗ = z ∗ = 0 . } is shown by the filled redrectangle. (a) Phase portrait for α = 1, which exhibits a symmetric boundary line given by the equation x = z (dashed line);(b) phase portrait for α = 10; (c) phase portrait for α = 200. For large growth rates α → ∞ , the boundary between theattraction basins tends to the line z = z ∗ . initial point of a trajectory, which was inside the attraction basin of the stable point for a first value of thegrowth rate, can happen to be found outside of it due to the change of the shape of the attraction basinfor a different value of the growth rate, or vice versa.ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December V.I. Yukalov, E.P. Yukalova, D. Sornette xz (a) xz (b) xz (c) Fig. 15. Phase portraits on the plane x − z for the parametric region D , with b = − . g = − 4, for different α .The dashed line shows the boundary between the attraction basins of the stable nodes { x ∗ = 0 . , z ∗ = 0 . } and { x ∗ = 0 . , z ∗ = 0 . } , which are represented by the filled green discs. The saddle point { x ∗ = 0 . , z ∗ = 0 . } is indicated by the filled red rectangle. (a) Phase portrait for α = 1; (b) phase portrait for α = 10; (c) phase portrait for α = 200. In the limit of large α ’s, the boundary between the attraction basins tends to the line z = z ∗ . t x(t)x ss (t) (a) t(b) z(t)z ss (t) t(c) x ss (t)x(t) t z(t) (d) z ss (t) Fig. 16. Temporal behavior of approximate solutions x ss ( t ) and z ss ( t ), obtained by the scale separation approach, as comparedto the exact solutions x ( t ) and z ( t ), for the case of mutualism, with the symbiotic parameters b = 0 . 15 and g = 0 . 7, with theinitial conditions { x = 2 , z = 6 } , for different growth rates α . The stable stationary state is { x ∗ = 1 . , z ∗ = 2 . } .(a) Population x ( t ) (solid line) and its approximation x ss ( t ) (dashed-dotted line) for α = 1; (b) population z ( t ) (solid line)and its approximation z ss ( t ) (dashed-dotted line) for α = 1; (c) population x ( t ) (solid line) and its approximation x ss ( t )(dashed-dotted line) for α = 10; (d) population z ( t ) (solid line) and its approximation z ss ( t ) (dashed-dotted line) for α = 10. We have illustrated this dynamic transition, caused by the distortion of the shape of the basin ofattraction, using a dynamical system describing the evolution of symbiotic species with different growthrates. The effect can happen under mutualism as well as under parasitism of the co-evolving species. Ashas been explained earlier [Yukalov et al. , 2012a,b, 2014a, 2015], the considered symbiotic equations cancharacterize various biological and social systems. Biological systems have also much in common witheconomical systems [Trenchard & Perc, 2016] as well as with structured human societies [Perc, 2016].ctober 7, 2018 15:42 Dynamic˙Symbiosis˙IJBC˙December REFERENCES t x ss (t) x(t) (a) t z ss (t)z(t) (b) t x(t)x ss (t) (c) t(d) z ss (t) z(t) Fig. 17. Temporal behavior of approximate solutions x ss ( t ) and z ss ( t ), obtained by the scale separation approach, as comparedto the exact solutions x ( t ) and z ( t ), for the case of parasitism, with the symbiotic parameters b = − g = − 2, with theinitial conditions x = 0 . z = 5, for different growth rates α . The stable stationary state is { x ∗ = 0 . , z ∗ = 0 . } .(a) Population x ( t ) (solid line) and its approximation x ss ( t ) (dashed-dotted line) for α = 1; (b) population z ( t ) (solid line)and its approximation z ss ( t ) (dashed-dotted line) for α = 1; (c) population x ( t ) (solid line) and its approximation x ss ( t )(dashed-dotted line) for α = 10; (d) population z ( t ) (solid line) and its approximation z ss ( t ) (dashed-dotted line) for α = 10. Therefore the described effect can occur in different nonlinear dynamical systems.As an example, where the described effect does happen in nature, it is possible to mention the ubiq-uitous symbiosis between fungi and plants. The proliferation of the Arbuscular Mycorrhizal fungi net-work at a late stage in plant life is well established to be beneficial for plant growth and reproduction[Kapulnik & Douds, 2000; Smith & Read, 2008]. 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