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Theoretical Population Biology 57, 5965 (2000) doi:10.1006tpbi.1999.1434, available online at http:www.idealibrary.com on Carrying Capacity and Demographic Stochasticity: Scaling Behavior of the Stochastic Logistic Model Jonathan Dushoff Institute of Physics, Academia Sinica, Nankang 105, Taipei, Taiwan Received March 18, 1999 The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R&1) 2 NR(R+1), where N is the habitat size and R is the basic reproductive number. ] 2000 Academic Press 1. INTRODUCTION biodiversity among competing species is maintained by a trade-off between competitive ability on one hand and reproductive and dispersal ability on the other. The same model was discussed, from the perspective of competing disease strains, by May and Nowak (1994). In this sort of hierarchical competition, the ``spacing'' between species at equilibrium may be determined by demographic stochasticity (Kinzig et al., 1999). Finally, the stochastic logistic model is equivalent to the stochastic SIS model in epidemiology, which was investigated in some detail by Na# sell (1995, 1996). Na# sell's results are similar in many ways to those presented here, but he did not focus on scaling based on habitat size (in the case of epidemiology, the habitat size is the population of available hosts), nor did he use the biased random walk heuristic. The stochastic logistic model investigated in this paper is the ``patch dynamic'' version, where the habitat is given an explicit size, and the only effect of density is that newborns cannot establish if they land on an occupied site. Historically, many authors (e.g., Pielou (1991)) have The stochastic logistic model is the simplest population model with discrete individuals and density dependence. Thus, it is the starting point for investigating the effects of demographic stochasticity in finite habitats. Many models of population viability are based on the stochastic logistic model, with added features such as population structure, Allee effects (Dennis, 1989; Gabriel and Burger, 1992; Stephan and Wissel, 1994) and environmental stochasticity (Leigh, 1981; Goodman, 1987). Furthermore, the stochastic logistic represents a good starting point for within-patch behavior of metapopulation persistence models, if it is thought that demographic stochasticity is an important component of persistence at the level of individual patches. The stochastic logistic model is also relevant to some models of biodiversity of competing species, and by extension models of speciesarea relationships. Tilman (1990) proposed a model, based on earlier work of Levins (Levins, 1969; Levins and Culver, 1971), where 59 0040-580900 K35.00 Copyright ] 2000 by Academic Press All rights of reproduction in any form reserved. 60 Jonathan Dushoff used ``stochastic logistic'' to refer to a somewhat broader model, where birth rates as well as death rates depend linearly on the number of adults present. The behavior of the two models should be similar. This paper presents an argument based on a biased random walk suggesting a scaling relationship between population size and reproductive number for the stochastic logistic model, together with evidence of the scaling relationship based on analysis and numerical simulation of the system. The scaling ideas presented here can be applied to a discrete-time version of this model as well, with some complications (Dushoff, submitted). Systems of this kind have a great deal in common with percolation models and other phase-transition systems that have been extensively studied by statistical physicists. The approach presented here is strongly analogous to what physicists call ``finite-size scaling'' of such models (see Stauffer and Aharony (1991, Chap. 4) for a review). Further use of techniques from statistical physics may have the potential to greatly increase our understanding of stochastic population models. 2. THE MODEL In the stochastic logistic model we consider a habitat with a finite number of sites N. Individuals die at rate m and give birth at rate f randomly into one of the N sites. If the site is already occupied, the newborn disappears. This is precisely the behavior of the dominant species in Tilman's competitioncolonization model (Tilman, 1990). The number of parameters in the model can be reduced by counting time in ``generations'' (units of 1m). Thus the density-independent birth rate becomes R= f m. Counting in terms of generations, we have the following instantaneous birth and death rates, b j and d j , for a population of size j : b j =Rj(1& jN ) d j = j. (1) To investigate persistence over long time periods, I will consider only populations whose basic reproductive number R is greater than 1. There are, however, interesting questions about time to extinction for populations with R1 as well. For R>1, define the carrying capacity K as the point at which birth rates are equal to death rates. The carrying capacity is given by K=(1&1R) N. (2) In the deterministic version of this model, the population would reach an equilibrium of K individuals. 3. A HEURISTIC SCALING ARGUMENT The key question is whether a population with birth and death rates given by (1) will persist. Of course, defining persistence is itself a tricky question. For the infinite-habitat version of (1), where b j becomes Rj, it is possible to ask simply whether a finite population will persist forever or eventually go extinct. If N is finite, however, the population will always eventually go extinct, although the mean time to extinction may be considerably longer than the expected lifetime of the universe. For a population that starts from a small number of individuals, we can heuristically break the question of persistence into two components. First, will the population establishthat is, reach or approach its carrying capacity? Second, how long can we expect the population to survive if it does succeed in establishing? For the question of establishment, we refer to the infinite system, which can be solved exactly (Karlin and Taylor, 1975). The probability of (infinite) persistence starting from j 0 individuals in an infinite habitat is P =1&1R j0. (3) If we think of the establishment question as representing the density-independent component, of demographic stochasticity, P makes sense as a heuristic estimate of establishment. To evaluate the probability of extinction of an established population, consider the population as performing a biased random walk. Whenever the population is less than the carrying capacity, the birth rate will be greater than the death rate; thus the random walk will be biased towards increasing population. In order to go extinct, the population must wander against this bias from the carrying capacity down to zero. The question is then how far a biased random walk is likely to stray against the direction of the bias. If the number of steps S is reasonably large we expect the distribution of positions of a biased random walk to be approximated by a normal distribution with mean &;S and variance S(1&; 2 ), where ; measures the amount of bias, defined here as the mean distance traveled per step. To approximate the stochastic logistic model as a biased random walk, we will look at the dynamics in terms of the number of births or deaths that have occurred (steps to the right and to the left), rather than 61 Scaling Behavior of the Stochastic Logistic Model in terms of time. It is straightforward to show that if the population size is j, the probability that the next step will be to the right is given by p j =b j (b j +d j ), and thus that the expected value of the next step is given by 2p j &1=(b j &d j )(b j +d j ). When j is zero, and there is no density dependence, and the mean distance traveled in one step is given by R&1 R+1 . Since the population will have to traverse the distance from the carrying capacity (where there is no bias) to zero in order to go extinct, we can assume, as a rough approximation, that the effective R&1 . mean bias is half of the bias at zero, so that ;r 2(R+1) 2 Assume that ; is small compared to 1, and imagine that we can estimate a quantity, :, that tells us how many standard deviations away from the mean our walk is likely to achieve in some long period of time. The standard deviation after S steps is approximately - S and the distance that the population can walk against the grain is given by : - S&;S. The maximum value of this quantity occurs when S=: 24; 2, giving a distance traveled of : 24;. At this point, we are not claiming to have any ideas about the value of :. Instead, the point of the heuristic argument is that we expect the characteristic wandering distance of the biased random walk to be on the order of 1;. The scaling argument of this paper is based on the hypothesis that the persistence of the population can be predicted by K;, which is the distance that the population needs to wander in order to go extinct, given in units of the characteristic wandering distance. If K is on the order of the characteristic wandering distance 1;, then we would expect the population to wander to extinction within some reasonable period of time, while if K is much larger than 1;, we would expect wandering to extinction to be quite unlikely and thus to take quite a long time. Hence, the hypothesis is that persistence of established populations will be predicted by U=K;= (R&1) 2 N. R(R+1) (4) In fact, since : 2 is proportional to U, the extinction probability or mean time to extinction might be expected to be exponential in U, if U is not too small. 4. METHODS It is straightforward to use a recursive algorithm to calculate the mean time to extinction for the system (1), starting from a single individual (Karlin and Taylor, 1975). The mean time to extinction is given by : j 1 k . ` 1& j k< j N \ + (5) Probability of persistence was calculated by directly integrating differential equations for the probability of the system (1) being in a given state, given that the population was not extinct. From the birth and death rates (1) we can write differential equations for the probability that there are exactly j individuals at a given time, f j : f4 j =b j&1 f j&1 +d j+1 f j+1 &(b j +d j ) f j . (6) The probability of the population being j increases with births from populations of j&1 and deaths from populations of j+1, and decreases with births and deaths out of populations of size j. The probability that the population has gone extinct by a given time is given by f 0 , and the probability that it has persisted, P, is given by 1& f 0 . We can therefore model the distribution of individuals, given that the population has survived, using p j = f j P. Modeling the p j instead of the f j improves numerical stability, and also allows integration to a steady state, after which it is possible to extrapolate exponentially to find the survival probability to any desired time. Results were checked by calculating the mean time to extinction from the integration results, and comparing with the analytic solution (5). 5. RESULTS Figure 1 (top) shows the mean time to extinction for populations in different-sized ``habitats'' (different values of N) plotted against the rescaled birth rate, R. As would be expected, if two populations have the same value of R, the one with a larger value of N will survive longer. In Fig. 1 (middle), the same values are plotted against carrying capacity K, rather than against the birth rate. It is useful to consider these two panels in terms of R and K. Figure 1 (top) shows that if we hold R constant, populations with a higher K survive longer, and Fig. 1 (middle) shows that if we hold K constant, populations with a higher R survive longer. In Fig. 1 (bottom) the same values are plotted yet again, this time against the quantity U derived from the random walk argument. U works surprisingly well as a predictor of mean time to extinction. Away from U=0, the line is also straight on a semilog plot, indicating that MTE is roughly exponential in U, as predicted. In fact, the scaling works too well to be explained by the heuristic 62 FIG. 1. Mean time to extinction starting from a single individual, plotted against reproductive number R, carrying capacity K, and the scaling parameter U. File: 653J 143404 . By:XX . Date:17:02:00 . Time:09:05 LOP8M. V8.B. Page 01:01 Codes: 1271 Signs: 334 . Length: 54 pic 0 pts, 227 mm Jonathan Dushoff FIG. 2. Mean time to extinction starting from the steady-state distribution, plotted against reproductive number R, carrying capacity K, and the scaling parameter U. 63 Scaling Behavior of the Stochastic Logistic Model argument alone; there must be some more detailed reason why this scaling works, which may be worth looking for. Figure 2 shows the mean time to extinction starting from the steady-state distribution. Once again the results are plotted first against R, then against K, and finally against U. Results are qualitatively similar to those in Fig. 1. Surprisingly, the scaling does not work as well as when we started from a single individual; this reinforces the suspicion that we do not understand everything about why the scaling works. Figures 1 and 2 show that the quantity U is useful in predicting the mean time to extinction of a model population with demographic stochasticity and density dependence. This is useful for understanding models of biodiversity. For models of single endangered species, however, we may be more interested in the probability of a population surviving for a fixed long time. Since extinctions from the steady-state distribution are distributed exponentially in time, we can calculate the probability of survival from the steady state to a fixed time T generations in terms of the mean time to extinction from the steady state M. The probability is exp(&TM ). Since Mrexp(U ), we can approximate the probability of survival to time T as FIG. 4. Probability of survival for 1000 generations, starting from a single individual, plotted against reproductive number R, together with the theoretical curve for a population in an infinite habitat to survive to time infinity. This is an S-shaped curve which is steep in the range where U is on the order of ln T. Figure 3 shows extinction probabilities from the steady state, together with this rule-of-thumb approximation. The question of survival from a small (non-steadystate) population is more difficult. As discussed above, the survival probability to time infinity for an infinite habitat (Eq. (3)) should provide an upper bound if T is large. Figure 4 shows the probability of survival for 1000 generations, starting from a single individual, together with the theoretical curve for an infinite habitat and infinite time. It is clear that the finite-habitat curves approach the theoretical curves as R becomes large. Based on the results for mean time to extinction, we might expect that the scaling quantity U would be a good predictor of the ratio between the observed survival probabilities and the theoretical value for an infinite habitat. Figure 5 shows the ratio PP plotted against U. Although U does a much better job of predicting the additional extinctions caused by a finite habitat than either R or K alone, it is clear that there is still room for FIG. 3. Probability of survival for 1000 generations, starting from the steady-state distribution, plotted against U, together with the rule of thumb (7). FIG. 5. Relative probability of survival for 1000 generations, PP , plotted against U, together with the approximation (7). Srexp(&T exp(&U )). (7) File: 653J 143405 . By:XX . Date:17:02:00 . Time:09:06 LOP8M. V8.B. Page 01:01 Codes: 4211 Signs: 2984 . Length: 54 pic 0 pts, 227 mm 64 improvement. Part of the problem is probably the fact that the expected relation between S and U (Eq. (7)) is rather sensitive to small errors in approximation. 6. DISCUSSION Mean time to extinction in a population following the stochastic logistic model (1) can be predicted by a single parameter, U, derived from a biased random walk heuristic argument. This parameter provides a way of linking spatial scale with the reproductive number of species whose persistence is governed by demographic stochasticity. Comparisons of predictions based on U with data for subpopulation persistence can provide evidence for the importance of demographic stochasticity in comparison with that of other factors such as trophic interactions, Allee effects, and environmental stochasticity. Although estimates of survival probability to a fixed time based on the scaling parameter U are not as precise as estimates of mean time to extinction, it is still possible to develop a rule of thumb for when a population can be expected to survive. For a population to be confidently expected to survive, starting from a steady-state distribution, we want to be beyond the steep part of the rule-ofthumb curve in Fig. 3, meaning that U should be greater than ln(T ). For the chosen time of 1000 generations, ln(T )r7. Estimates of survival probability from a small population are even less precise, but we can extend the rule of thumb above. For a population to survive, it should be reasonably protected from both wandering to extinction from the steady state, and from failure to establish at all. Thus we want U to be large compared to ln T, as before, and we also want R j0 to be small, meaning that j 0 should be large compared to 1(R&1). Note that these estimates apply only to effects due to stochasticity in birth and death rates. For many species, environmental stochasticity may also be important, which could sharply reduce persistence estimates. For monoecious, sexual species, it is known that effects such as failure to find mates and unfavorable sex ratios are important for small populations (Dennis, 1989; Gabriel and Burger, 1992). This problem can be partially corrected in this model by counting only females (and thus halving the habitat size), but this is only a rough approximation to a full sexual model. The scaling result also has some implications for models of biodiversity of competing species. If the longterm survival probability is closely related to the quantity UrN$ 2, then as the number of sites, N, increases, the Jonathan Dushoff amount of ``breathing space'' by which R must exceed one for the species to persist decreases at the rate 1- N. Hence in species diversity models based on the paradigm of Tilman (1990) we would expect the number of species found in the long run to increase roughly as - N. This prediction provides a null model for speciesarea relationships under Tilman's paradigm, under the assumption that extinction rate is determined primarily by environmental stochasticity (rather than disease or environmental stochasticity, for example) and that spatial structure may be ignored. Tilman's paradigm postulates an absolute competitive hierarchy. It can usefully be seen as one end of a spectrum of theories of diversity of competing species, where Hubbell's (1997) models of identical species are the other end of the spectrum. It should be pointed out that both Leigh (1981) and Na# sell (1995) have already shown that N$ 2 is a useful scaling quantity for this model, although neither one found the further correction embodied in (4), which seems to substantially improve results at this scale. The work of Leigh also provides insight into why the exponential approximation may work better for populations that start from a single individual than those that start from carrying capacity, or from the equilibrium distribution. Future work may be able to incorporate his ideas into the framework provided here to improve results for populations starting from the equilibrium distribution. Both works contain more complicated approximations which in fact work better than the approximation given here. It is hoped that the simple scaling relation here will prove useful both in assessing the relative importance of demographic stochasticity compared with other factors contributing to local or global extinction in the field, and in assisting with the analysis of more complex and realistic models, which may have similar underlying structure and scaling rules. It is further hoped that the biased random walk heuristic will prove useful in understanding structurally more complex models, in particular ``SIR'' disease models, which consider the possibility of acquired immunity to disease. REFERENCES Dennis, B. 1989. Allee effects: Population growth, critical density and the chance of extinction, Nat. Resour. Modeling 3, 481538. Dushoff, J. ``Demographic Stochasticity in Discrete-Time Models: A Scaling Result,'' submitted. Gabriel, W., and Burger, R. 1992. Survival of small populations under demographic stochasticity, Theor. Popul. Biol. 41, 4471. 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Regional coexistence of species and 65 competition between rare species, Proc. Nat. Acad. Sci. U.S.A. 68, 12461248. May, R. M., and Nowak, M. A. 1994. Superinfection, metapopulation dynamics and the evolution of diversity, J. Theor. Biol. 170, 95114. Na# sell, I. 1995. The threshold concept in stochastic epidemic and endemic models, in ``Epidemic Models: Their Structure and Relation to Data'' (D. Mollison, Ed.), Cambridge Univ. Press, Cambridge, UK. Na# sell, I. 1996. The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Probabil. 28, 895932. Pielou, E. C. 1991. ``Mathematical Ecology,'' Wiley, New York. Stauffer, D., and Aharony, A. 1991. ``Introduction to Percolation Theory,'' 2nd ed., Taylor 6 Francis, London. Stephan, T., and Wissel, C. 1994. Stochastic extinction models discrete in time, Ecol. Modelling 75 76, 183192. Tilman, D. 1990. Competition and biodiversity in spatially structured habitats, Ecology 75, 216.