A simple individual-based population growth model with limited resources
AA simple individual-based population growthmodel with limited resources
Luis R. T. Neves a, ∗ , Leonardo Paulo Maia a a Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, S˜ao Carlos, Brazil
Keywords: population dynamics; logistic growth; individual-based modelling.
Abstract:
We address a novel approach for stochastic individual-based modelling ofa single species population. Individuals are distinguished by their remaininglifetimes, which are regulated by the interplay between the inexorable running oftime and the individual’s nourishment history. A food-limited environment inducesintraspecific competition and henceforth the carrying capacity of the medium maybe finite, often emulating the qualitative features of logistic growth. Inherentlynon-logistic behavior is also obtained by suitable change of the few parametersinvolved, composing a wide variety of dynamical features. Some analytical resultsare obtained. Beyond the rich phenomenology observed, we expect that possiblemodifications of our model may account for an even broader scope of collectivepopulation growth phenomena.
1. Introduction
Historically, population dynamics has acquired a much broader perspective along the development of math-ematical modeling, playing a central role in the study of the spreading of epidemics, opinions, computer virusesand even fake news or characterizing the evolution of languages and cancer cell populations [1]. Notwithstandingthe inherently nonlinear character of even the simplest interactions among individuals, a widespread practicewhen studying populations consists in building dynamical models by adding simple mathematical expressionsthought of as ingredients, each corresponding to a basic populational behavior. Roughly speaking, one expectsthe full model to keep the elementary ingredients but also being able to allow the emergence of more complexbehaviors through nonlinearity. Aside the basic linear models for either growth or decay of a set of independentindividuals, the simplest model for population dynamics is the logistic interaction among its members.Within the scope of single-species modelling – to which this manuscript is entirely dedicated –, there aretwo well-stablished deterministic logistic models, distinguished in the nature of their temporal evolution [2,3]. Verhulst’s exactly solvable continuous time logistic equation, ˙ x = λx (1 − x/K ), only presents monotonictrajectories of the total population x ( t ) > x (0) towards the carrying capacity K >
0. Far from this condition, it behaves roughly as an exponential, either decaying when x ( t ) >> K or growing with rate λ > < x ( t ) << K . The second deterministic paradigm, the logistic map, x t +1 = λx t (1 − x t ), is a discrete time dynamical system with a rich variety of behaviors regulated by thebifurcation parameter λ >
0. It can converge to a fixed point but its asymptotic behavior can also consistof oscillations and it is even capable of achieving chaotic behavior through a period-doubling route. Despitenot being a proper discretization of the continuous-time version, such an extensive phenomenology renderedpopularity [4] to the logistic map as an example of generation of complexity by simple models.Regardless of their profound conceptual, mathematical and phenomenological distinctions, the Verhulstequation and the logistic map share at least two important, general features that may put them in the sameclass if we are to characterize population dynamics models. First, the only relevant variable concerning thestate of the population is its size (or density). In other words, the population is completely described by thenumber of living individuals and, as a consequence, the dynamics of its temporal evolution can in no waydepend on its internal structure or distinction among individuals. It is therefore assumed that the totality ofthe complex interactions amongst individuals and between population and environment may be encoded just inhow the employed model maps the population size into its temporal evolution, in a kind of “mean-field” fashion. ∗ Email address: [email protected] (Luis R. T. Neves ) a r X i v : . [ q - b i o . P E ] J un ollowing the literature, we shall refer to any model of this kind as a population-level (or population-based)model.A second, more obvious aspect is that both the logistic ODE and logistic map are deterministic dynamicalsystems: specifying the state of the system in a given instant of time suffices, at least in principle, to predictthe exact outcome of an arbitrary time evolution.Other notorious, although more specific models that fit within the same class are the Monod and Droopmodels for microbial growth in laboratory conditions [5, 6]. These models may include other variables than thepopulation size, such as nutrient concentration in the medium; however, when it comes to the description of thepopulation itself, only its density is taken into account.Deterministic, population-based models are of acknowledged historical and practical importance. Neverthe-less, since the 90’s a notorious growth has been reported in the employment of a radically different approachwhich we may generically name individual-based modeling [7, 8]. In this framework, rather than writing downsome equation determining the time evolution of the population size as a function of it (and possibly somepopulation-external variables), the starting point is the description of the elementary units of the system, i. e.,individuals. The modeller then designs some idealized behavior rule which governs the actions of this singleindividual, and expects the global system to follow some ecologically reasonable law. Besides, the mentionedindividual-level behavior rules are typically of probabilistic nature. Such rules may try to capture real actionsperformed by individuals, such as feeding, replicating, dying, or moving. We shall label any of these models anindividual-based model (IBM). We ought to make it clear that many authors distinguish between individual-based and agent-based models [7]; we choose to ignore such distinction and just employ the term IBM to meanthat the fundamental dynamical laws are formulated at the level of the individual, rather than that of the wholepopulation.The fact that IBMs allow us to unveil “macroscopic” collective behavior as emerging from “microscopic” in-dividual one (and, sure enough, from the intricate network of interactions that takes place) is itself a tremendousadvantage from the theoretical point of view; besides, there is the obvious fact that real populations do exhibitobservable variations from one individual to another, which cannot be captured by population-level models[9, 7]. This approach has also shown great practical importance, providing much more accurate predictionson real ecological situations, also opening up the possibility of learning about the behavior of individuals fromobservational data concerning the dynamics of the entire population [8].In this work, we do not discuss models which consider the spatial distribution of individuals; that is, nonotion of space is explicitly taken into account. We shall refer to such restrict models as population growth models, while the more general term population dynamics might include spatial issues.Some of the so-called logistic IBMs are designed in such a way as to recover the logistic growth curvein average, or in some deterministic limiting case; this, however, is often achieved through the imposition ofindividual behavior rules that somehow mimic the mathematical structure of Verhulst’s equation – for instance,the probability of an individual replicating in a short interval of time being proportional to 1 − N/K , where N is the size of the population [9, 10, 11]. Though useful and convenient, this approach is artificial in thesense that such probabilistic law is largely an ad hoc construction, and the very meaning of the parameter K ,for example, is clear only as long as the population-level model that gave birth to this individual-based one isknown a priori . In other words, it is difficult to make sense of such a law in the individual level by itself.So motivated, one of the goals of this work is to introduce an IBM based upon a simple, intuitively-justifiedset of individual behavior rules which might reproduce a logistic-like curve as a genuinely emergent phenomenon.The same model, nonetheless, will give rise to many other different qualitative dynamical signatures, such asdamped oscillation towards equilibrium, and even perpetual growth. The model is based upon the descriptionof individuals that can reproduce asexually and indirectly compete for nourishment resources. We model aresource-limited environment, and the available food is distributed randomly among individuals; the finitenessof this food is what induces the referred competition effect and, consequently, may stop the vegetative growthof the population. Only three fixed parameters rule individual behavior and, thence, determine the differentcollective growth patterns that emerge. We have been able to derive few analytical results, from which wehighlight the equation for the carrying capacity of the medium as a function of the referred individual-levelparameters. Due to the complexity of the system, however, much of the carried out characterization of thedynamics was based in the outcome of numerical simulations.We now outline the structure of this manuscript. In § § § § § § § §
2. The model2.1. Definition
We start with a qualitative overview of the individual behavior rules. Individuals are modeled as beingdistinguishable upon a single, dynamical attribute: their remaining lifespan ( τ ), which naturally decreases atconstant pace as time runs. If an individual reaches τ = 0, it is regarded as dead. Feeding is the only mechanismthat may decelerate such process, and the (limited) available food per time step is randomly distributed. Ifthe constant amount of food to be delivered at a given instant exceeds the present size of the population, noaccumulation effect occurs; the excess is simply discarded. Finally, reproduction is asexual and at constant rate;newborns are set at a fixed value of τ .Formally, our population at a given instant t is a list { τ i } i =1 ,...,N ( t ) of integers τ i (to be thought of as theremaining lifetime of the i -th individual), whose time evolution is defined as the sequence of the following threeoperations:1. Feeding: if N (cid:54) α , we make τ i ← τ i + 1 , ∀ i . Elsewise, α integers in { , ..., N } are drawn at random, and τ i ← τ i + 1 for the drawn values of i ;2. Aging: for every i , τ i ← τ i −
1. If τ i = 0 for some i , such entry is deleted from the list;3. Reproduction: for every individual in the population, there is a probability ρ that a new individual will beborn, at τ = ω .Henceforth, α, ω ∈ N and ρ ∈ (0 ,
1) are our fixed parameters. Formally, of course, after death and birthoperations one should rearrange the indexes i so as to remove the “empty” entries etc. By convention, the threemanipulations above constitute one time step, and, in general, we describe the state of the population between time steps. Although so defined, describing our population as a list of τ i ’s is unnecessarily complicated (for instance,the very “shape” of the list, N ( t ), is itself a dynamical variable). For, steps 1 and 2 above assure that a givenindividual’s τ value never increases throughout a time step – it either remains constant or decreases in a unit–; besides, individuals are always born at τ = ω . Thus, if we assure an initial condition such that τ i (cid:54) ω , ∀ i ,which we shall always do, it is certain that such constraint will remain true at any instant of time. Now we areallowed to partition our population with respect to the lifespans, writing down a list X t = (cid:0) X t , ..., X ωt (cid:1) , (1)where X jt is the number of individuals for which τ = j at time t . The obvious advantage of this description isthat now the dimension of the list (which specifies the state) is a fixed parameter, namely ω . Our main goal isthen to describe the mathematical features of the dynamics induced by steps 1-3 on the state X t .A straightforward translation of the individual-based dynamics into the terms of this class-structured de-scription leads to the equations below, which encode all the information about the time evolution of X t : X it +1 = (cid:40) ( X i +1 t − Q i +1 t ) + Q it , i = 1 , ..., ω − Q it + R t , i = ω, (2)where Q it is the number of fed i − th class individuals in the step t → t + 1, and R t the total of newbornsgenerated in the same step. We shall always denote random variables by capital Roman letters and fixedparameters by lowercase Greek letters. 3 . Results3.1. Equilibrium. Subcritical growth The stochastic dynamics is extremely simple so long as N t ≡ (cid:80) i X it (cid:54) α , for in such case Eqs. (2) reduceto X it +1 = X it , i < ω and X ωt +1 = X ωt + R t ; besides, R t is binomially distributed with parameters ( N t , ρ ). Thus N t grows exponentially in average and no steady state could be reached. We shall call this abundance regime ,in opposition to the competition regime ( N t > α ); the latter exhibits a much richer dynamics, with which weshall be mainly concerned.From Eqs. (2) it is easy to derive the expectation value of X t +1 , conditioned to X t = x ≡ ( x , ..., x ω ) (i. e.,for fixed X t ). We can show (Appendix Appendix A) that, under competition regime, E x (cid:0) X it +1 (cid:1) = (cid:40) ( α/n ) x i + (1 − α/n ) x i +1 , i < ω ;( α/n ) x i + ρ (cid:2) n − (1 − α/n ) x (cid:3) , i = ω, (3)where the subscript x stands for the referred conditioning.We shall adopt a simple definition of stochastic equilibrium, in that a state x ∗ is said to be of equilibriumif it satisfies E x ∗ ( X t +1 ) = x ∗ , (4)that is, the condition X t = x ∗ is stationary in average . Combining condition above with Eq. (3) yields x ∗ = ( n ∗ /ω, ..., n ∗ /ω ) , (5)where n ∗ ≡ (1 + ρ ) α − ( ω − ρ (6)is the equilibrium size of the popoulation, analogous to the carrying capacity in the logistic model. In particular,Eq. (5) shows that, in equilibrium, all “classess” are equally populated. It is immediately noted that Eq. (6)is potentially pathological in that, if 1 − ( ω − ρ (cid:54)
0, it could not possibly represent the size of a population.Indeed, the very definition of our population (along with its individual behavior rules) cannot make sense ofsuch thing as negative population size; thus, what Eq. (6) suggests is that, for a certain regime of parameters,no steady state can hold whatsoever. For that reason we shall call conditions 1 − ( ω − ρ >
0, = 0 and < subcritical , critical and supercritical regimes, respectively.Analyzing the subcritical case as our starting point, we show that numerical simulations reinforce ouranalytical predictions. (Initial condition is always X = (0 , ..., , N ), unless otherwise stated.) In Fig. 1,typical behavior of N t in subcritical regime is depicted, for two different parameter choices. Population sizeindeed reaches the steady-state value given by Eq. 6.As it turns out, oscillations in population size are not only of stochastic nature (as seen in a single realization),rather surviving to the averaging of many simulations, as one sees in Fig. 1 (a). Such oscillations, however,do not always take place, even in the subcritical case; Fig. 1 (b) shows instead a monotonous growth towardequilibrium. Actually, it was verified that the occurrence of oscillatory behavior may depend even upon theinitial conditions; we shall return to this point in § N t alone does not tell much about the dynamics of X t , which has to be visualizedat any given instant as a histogram. Fig. 2 shows, for the same dataset of Fig. 1 (a), the state X t for somevalues of t . It is shown that equilibrium state given by Eqs. (5) and (6) is indeed reached for large values of t . Inspired by Eq. (3), one may define a deterministic map that partially captures the features of our original,stochastic system. Specifically, define a state y t = ( y t , ..., y ωt ) which evolves according to the law y it +1 = (cid:40) ( α/n t ) y it + (1 − α/n t ) y i +1 t , i < ω ;( α/n t ) y it + ρ (cid:2) n t − (1 − α/n t ) y t (cid:3) , i = ω, (7)4igure 1: Simulation results in subcritical regime.Figure 2: Evolution of X t for the same data set of Fig. 1 (a). Red, dashed lines assign the predicted equilibriumvalue of X i , namely n ∗ /ω . 5igure 3: Comparison between average of stochastic dynamics and deterministic map, in both a case wheretransient fluctuations are in good agreement and another where they are not.Figure 4: Simulation results for a supercritical choice of parameters. Exponential fit was obtained by linearregression on log (cid:104) N t (cid:105) , with time ranges previously chosen so as to disregard transient growth. The obtainedangular coefficient was a = 1 . × − with determination coefficient R = 0 . n t ≡ (cid:80) i y it > α , and y it +1 = y it + δ i,ω ρn t elsewise. Such dynamics is by construction identical to the averagedresult of that of Eqs. (2) only in abundance regime or in steady state; how close the two are in any otherdynamical condition is, a priori , a question to be answered numerically, given our lack of stronger analyticalresults for both systems. It turns out that the deterministic map nicely emulates the stochastic one for somechoices of the parameters, while, for others, the transient fluctuations widely differ in amplitude (see Fig. 3). As seen in § ω − ρ = 1 or ( ω − ρ >
1. Simulations revealed thatin the latter case (supercritical) a transient growth is followed by steady, exponential growth, as illustrated inFig. 4. Other choices of parameters in the same region have always exhibited the same behavior.In its turn, critical parameter regime exhibits linear growth following the transient. By means of an ansatz motivated by such numerical observation, we have successfully calculated the asymptotic time dependence ofdeterministic y t in this regime, namely y it = 2 ραω t + 2 ραω i + b , (8)whence the population size n t = (cid:80) i y it follows 6igure 5: Simulation results for population growth in critical case. The three straight lines (simulation average,deterministic map and adjusted line) are virtually undistinguishable. The slope obtained by linear regressionwas 0 .
995 with determination coefficient R = 0 . ρα = 1.Error bars have been omitted for being essentially of the width of the lines. n t = 2 ραt + ρα ( ω + 1) + ωb , (9) b being an undetermined constant (see Appendix Appendix B for proof). Simulations confirm our predictions,besides showing that the behavior of E ( X t ) is finely captured by the deterministic map in critical case (see Figs.5 and 6). We have verified previously ( § N ) alone, we can control such property for both sets of parameters simulated in Fig. 1. Fig. 7(a) also shows that these oscillations may be present in the deterministic map even when the averaging of manyrealizations of the stochastic model shows monotonous growth; Fig. 7 (b), on the other hand, is an example ofgood agreement between both curves.As we are not concerned with the characterization of transient growth in critical and supercritical regimes,there is nothing to be discussed about the role of initial conditions in those cases.
4. Discussion In § § X t for the same simulation data set of Fig.5 at different times. In each case, a straight line was fitted by linear regression. The numerical outputs were:(a) a = 5 . × − , R = 0 . a = 4 . × − , R = 0 . a = 4 . × − , R = 0 . a = 4 . × − , R = 0 . a = 2 ρα/ω ≈ . × − . Notshown, deterministic map exhibits an even finer adjustment to the expected slope.Figure 7: Simulation results for same choices of parameters as in Fig. 1, but different initial sizes, inducingqualitatively diverse transient dynamics (as to the presence or absence of damped oscillations).8f much simpler nature and interpretation is the growth pattern depicted in Figs. 1 (b) and 7 (a), where earlyexponential growth is followed by an inversion of the concavity and a steady approach to the carrying capacity.This is in qualitative correspondence to the paradigmatic logistic growth, broadly endorsed by empirical evidenceconcerning populations of species as diverse as flies [14], protozoans [14], and humans [15]. Our model, thus, maybe regarded as capable of making logistic-like growth emerge from simple, biologically reasonable individual-level behavior rules. It is noteworthy that many of the so-called individual-based logistic models achieve thedesired population-level behavior only through the arbitrary imposition of less natural rules, such as birth anddeath probabilities depending on population density in a carefully chosen fashion that is indeed reminiscent ofthe logistic equation. Examples of works that include this approach in more or less central positions are Refs.[9, 10, 11].Still in the case of subcritical regime, one important signature of our model is the expression for n ∗ , thecarrying capacity of the environment, in terms of the fixed parameters. Within this region of the parameterspace, n ∗ is an increasing function of ρ, α and ω , as expected. Moreover, the dependency on α , the onlyeasily-controlled parameter in a hypothetical laboratory situation, is as simple as it could be.We now attempt to discuss and interpret the intriguing case of supercritical growth. Sure enough, oneshould be careful in trying to make sense of such thing as unlimited, exponential growth under resource-limitedenvironmental conditions. Strictly speaking, this is undoubtedly a signature of a rather artificial feature of ourmodel, namely the fact that individuals are always born with a predetermined (in this case, presumably long)“life expectancy”, despite of their parent’s. Indeed, we are forced to acknowledge that in a sense this is a borderof the scope of validity of the model. However, we can reinterpret the supercritical growth as being meaningful(i. e., still eligible to describe some real phenomenon) only for a finite range of time, much like the Malthusianvegetative growth model ˙ y = ρy may be valid only as long as the population size y is small enough.A natural objection arises: is our model, in the supercritical regime, merely reproducing the same infamousperpetual exponential growth just in an infinitely more complicated manner than the oldest and simplestmathematical model of population dynamics does? Not quite, as we shall argue. First, we recall that earlyexponential growth, conditioned to the abundance of food, is always verified in our model, regardless of theparameter regime, as we have verified even analytically in § another period of exponentialgrowth may be achieved, even away from the abundance regime (understood in the precise sense of § has to be radically different. For, in such setting, the size of the population may beorders of magnitude larger than the available food per time step (see Fig. 4); clearly, for large enough values of N , a typical individual will never get fed at all in its lifetime. Such individual, then, will die after a time intervalof length ω , generating an average of ρω offsprings in the meantime. Thus, the net contribution of this trackedindividual to the population size, ω time steps after its birth, is ρω −
1; the supercritical condition ρ ( ω − > ρω − > ρ >
0, elucidating why, despite the virtual absence of food, this combination of large birthrate and life span assures a sustained growth of the population. Of course, unveiling such a mechanism is in noway a validation of the model in the referred regime; it remains an open task to find a real-life situation thatmight be modelled by it.Finally, critical growth is the most delicate case. It has the same undesirable feature of indefinite growth asdoes the supercritical regime; in qualitative terms, what differs one from the other is that, in the supercriticalcase, the dependence of (cid:104) N t (cid:105) on t appears linear only for a limited lapse of time, subsequently revealing itshigher order corrections, only adequately described by an exponential function. In the critical case, for its turn,the time for those corrections to become evident may be regarded as infinite – the linear behavior remainsunchanged. It can also be thought of as a limiting case of subcritical growth, in which the time it takes forequilibrium to be reached is infinitely long. It is thus clear that this regime of growth is by its nature only aborder case between two more robust dynamical settings, so that trying to find an independent correspondencebetween it and a real life situation must be even tougher than in the supercritical case. On the other hand,from the point of view of dynamical systems, this kind of transition phenomenon might be of interest.
5. Conclusion
In summary, we have seen that an interesting diversity of dynamical patterns appear in our model, with higheror lower degrees of connection to ecological reality. The most important message is that all of them emerge froma single, biologically reasonable set of individual behavior rules, in which one attempts to model the elementaryfeatures of food-limitation-induced intraspecific competition. Moreover, the transition among those different9rowth patterns is ruled, aside from initial conditions, by only three parameters – all of straightforward biologicalinterpretation: birth rate (the inverse of average generation time), available food per unit time (a measure ofthe environment’s resource abundance) and the lifespan of a newborn in scarcity of food. In particular, in theecologically appealing subcritical case, we have derived an expression for the environment carrying capacity asa function of those parameters (Eq. 6). All of this may indicate that a rich, useful, novel class of populationgrowth models may be built upon the seminal ideas here addressed, by introducing punctual modifications inthe behavior rules that we have defined. Such adaptations might account for phenomena not captured by ourmodel, such as extinction, inheritance, and even evolution.
Acknowledgements
L.R.T.N. thanks his fellow and friend Solano E. S. Fel´ıcio for important discussions carried out along thedevelopment of the research here addressed. L.R.T.N.’s work has been supported by the S˜ao Paulo ResearchFoundation (FAPESP/Brazil) under grant no. 2018/13032-1.
Appendix A. Transition expectations
Taking the expected value in both sides of Eqs. (2), conditioned to X t = x , we obtain E x (cid:0) X it +1 (cid:1) = (cid:40) x i +1 + E x (cid:0) Q it (cid:1) − E x (cid:0) Q i +1 t (cid:1) , i < ω ; E x (cid:0) Q it (cid:1) + E x ( R t ) , i = ω. (A.1)As we are concerned only with competition regime, each Q it is hypergeometrically distributed with parameters n, α, x i , whence E x (cid:0) Q it (cid:1) = ( α/n ) x i . In its turn, R t has binomial distribution once the Q it ‘s have been realized,say Q it = q i ; for, x − q individuals will have died, the remaining n − ( x − q ) being able to generate a newbornwith probability ρ . Thus, E x (cid:0) R t | Q t = q (cid:1) = ρ (cid:2) n − ( x − q ) (cid:3) (A.2)and, by the Total Probability Theorem, E x ( R t ) = (cid:88) q E x (cid:0) R t | Q t = q (cid:1) P x (cid:0) Q t = q (cid:1) (A.3)Using Eq. (A.2) in Eq. (A.3) and exploring the linearity of E x (cid:0) R t | Q t = q (cid:1) will give raise to two terms,one being the mean of Q t and the other, simply its normalization. Taking this result into Eq. (A.1) immediatelyyields Eq. (3). Appendix B. Analytic solution for critical growth
Motivated by numerical results, we try in Eqs. 7 a solution in which each y it grows linearly with time, with therestriction that all the ω angular coefficients be equal. That is, we try y it = at + b i ; thus, n t = ωat + b, b ≡ (cid:80) i b i .Taking the form of y it into the first of Eqs. (7) gives b i +1 − b i = a − α/n t . (B.1)In an exact sense, Eq. (B.1) implies that no such solution may exist, for the left-hand side is a constant,whereas the right-hand depends upon t (we have seen in § n t cannot be stationary in critical regime).Nevertheless, since n t grows indefinitely with time in our ansatz , we may assume that our solution only holdsasymptotically for large values of t , so one may approximate 1 − α/n t ≈ b i +1 = a + b i , i = 0 , ..., ω −
1. We thus write b i = b + ai and b = ωb + aω ( ω + 1) / y it and n t along with the obtained form of b i and b in the second of Eqs.(7), also imposing the critical relation ρ ( ω −
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