Elsevier

Ecological Modelling

Volume 212, Issues 3–4, 10 April 2008, Pages 334-341
Ecological Modelling

The effects of endogenous ecological memory on population stability and resilience in a variable environment

https://doi.org/10.1016/j.ecolmodel.2007.11.005Get rights and content

Abstract

Endogenous ecological memory (EEM) refers to the phenomenon where past states of a system (i.e. population density) can influence present states. In this paper, we address the effects of EEM on qualitative changes in population dynamics over time. The goal was to determine the effects of EEM on the stability and resilience of a population. Since natural populations do not live in isolation, we analyze the discrete-time Ricker model St+1 = St exp (R) exp (−RSt/K) with environmental stochasticity. We explore the model when time-delays are restricted to the density-dependent exp(−RSt/K) component of the model and lognormal multiplicative noise is incorporated as well. Simulation results show that population stability, as measured by the standard deviation of population density about the carrying capacity, decreases as the duration of EEM increases. We observe boom–bust cycles in population density for significantly long time-delays in density-dependence. These cycles become more extreme as the regulating effects of density-dependence are increasingly delayed, resulting in decreased population stability. Simulation results also show that population resilience, as measured by the return time to equilibrium after perturbation (Tr), decrease with increasing fecundity R in a pattern that differs from the predictions of some theoretical work where memory is absent. Finally, the presence of environmental stochasticity changes the resilience properties of the model, relative to the equivalent noise-free model.

Introduction

The literature is rich with examples of discrete and continuous population dynamic models that explore the effects of time-delays in the model(s) variables on qualitative and quantitative properties of population dynamics (see Halanay, 1966 and May, 1973). For example, Maly (1978) used a predator–prey model (Maly, 1975) to explore qualitative and quantitative prosperities of population stability. The model demonstrated that time-delays in changed food supply (Paramecium caudatum) of the predator (Didinium nasutumum) decreased stability marginally. Nunney (1985) explored the stability properties of a single population model and a predator–prey model. For the single-species model, the study demonstrated that short delays in maturation time produced a variety of stability regimes, including extreme decreases and increases in stability (e.g., boom–bust cycles). For the predator–prey model, short delays in prey maturation in the presence of recruitment acted to stabilize the prey population about its regulatory equilibrium. However, short delays in maturation time markedly reduced stability. Conversely, when prey recruitment was non-regulatory around the equilibrium, short delays in prey maturation acted to stabilize the qualitatively unstable system. These results confirmed earlier work by Hastings (1983), and were robust to changes in initial conditions. Forchhammer et al. (1998) developed a model to explore data from a plant–deer–climate system. Statistical analysis of the model demonstrated that direct and delayed density-dependence in deer abundance over time had a significant effect on the development of deer populations. The study also demonstrated that time-delays in climate variability predicted deer abundance; two-year delays in warm, snowy winters had resulted in increased deer abundance. Benton et al. (2001) attribute the lag in density-dependence to maternal effects and showed that a lag generally increased population variability and generally destabilize dynamics but suggested that the effects of noise could actually restabilize dynamics.

In this paper, we explore the concept of ‘ecological memory’ in terms of time-delays, and use it to investigate the qualitative dynamics generated by a simple ecological model. Many published models that explore the effects of time-delays on population dynamics fit within the ecological memory framework presented here. Features of our approach that are frequently lacking from studies of the effects of ecological memory on population dynamics include: the ability to define ecological memory in a mathematically clear and compact manner (i.e. in terms of time-delays), and a framework for exploring the effects of time-delays for a large class of mathematical models, each of which can utilize empirical data from studies to determine the effects of time-delays on ecological populations in situ. Thus, the ecological memory concept described herein may provide an inferential framework for fitting population dynamic models, given observational data.

Section snippets

Ecological memory

Implicit in ecology is the idea that the past state of a population within an ecosystem can influence present or future dynamics of populations within that ecosystem (e.g., relict woodland species in grasslands) (Margalef, 1961, Warner and Chesson, 1985). This concept leads to the idea of “ecological memory” (hereafter, EM), which Padisak (1992) described as “the capacity of the past states or experiences to influence present or future responses of the community” (p. 225). This study concluded

Mathematical models for ecological memory

Historically, mathematical analysis of the ecological dynamics of an organism within an ecosystem has begun with mathematical models that explore the growth dynamics of populations. The most commonly used models have been those based on the discrete-time logistic model for the growth of a population (Verhulst, 1838), which is described by the following equationSt+1=RSt1StK,S00,R>0,K>0,where K > 0 describes population carrying capacity, R > 0 is the population fecundity, and the population size

Environmental stochasticity

Ecosystems are not closed systems; hence, studies of the effects of EEM on population dynamics necessarily require the inclusion of environmental effects. At short time scales, unpredictable climate changes give rise to environmental stochasticity. For example, short-term changes in temperature can exhibit irregular patterns that can be described in terms of stochastic fluctuations (Clifford and McClatchey, 1996). In Eq. (5), let the state at time t be a function of environmental stochasticity,

The stability–R and resilience–R relationship

In order to analyze the effects of EEM on population stability and resilience in the presence of environmental stochasticity, we expand upon past theoretical work that shows a close relationship between population fecundity (R) and both stability and resilience. Naturally, according to these definitions, populations can be said to be more or less stable and more or less resilient, depending on parameter values. Xu and Li (2002) found that for models similar to Eq. (6), but without EEM,

Measuring stability and resilience in a stochastic system

In order to measure population stability, we computed the average standard deviation (square of the average variance, hereafter, average standard deviation (ASD)) of population density at each point in the time-series about the corresponding equilibrium of the noise-free model at the same time point, summed over all time points. A smaller ASD indicates that the population is more stable, whereas a larger ASD value indicates a less stable population. Thus, we assume that the ASD is a suitable

Measuring resilience

Numerous studies in the ecological literature show that the return time to the equilibrium after perturbation for models similar to Eq. (6) in the absence of EEM and environmental stochasticity is given by the formula Tr  1/R (May, 1981, Pimm, 1991, Stone et al., 1996). No studies, to our knowledge have examined the effects of EEM and environmental stochasticity on return time (resilience). In order to measure population resilience, we used numerical simulations of Eq. (6) to determine the time

The effects of EEM on population stability

Numerical simulations of Eq. (6) show that for increasing population fecundity R, the duration of EEM (amount of lag) has profound effects on population stability. Fig. 2 shows representative time-series for various values of R and lag. An increase in R and/or lag leads from small-magnitude fluctuations to increasingly large oscillations and possibly boom–bust cycles. When the duration of EEM is greater than or equal to 1 and R  1, there is a qualitative shift from stable population dynamics to

The effects of EEM on population resilience

Fig. 4 shows two examples of how the population described by Eq. (6) responds to a sudden environmental perturbation when the delay in population regulation is lag-2 for R = 0.2 and R = 0.8. It is clear that the population is more resilient (smaller return time) when R = 0.2 than when R = 0.8. Fig. 5 shows that for lag-1 in Eq. (6), the resilience–R relationship obtained by using the definition of resilience in the present work is similar to results obtained by Stone et al. (1996). However, unlike

Biological implications

The effects of EEM on population stability and resilience are not only determined by the deterministic dynamics of population regulation, but are also dependent on stochastic uncertainties faced by the population described by Eq. (6). Results generated from our analysis show that in the presence of EEM, the population reacts differently because of the stability–R and resilience–R relationship. For some values of R, populations are highly sensitive to increases in delays in population regulation

Conclusions and discussion

In this paper, we extended the definition of ecological memory used in past theoretical and empirical work by examining the effects of EEM on population stability and resilience in the presence of environmental stochasticity. Although previous studies have studied the dynamics of Ricker-type models in the presence of noise, this is the first study to our knowledge that also incorporates EEM in such models in an ecological context. This study specifically explores these effects based on the

Acknowledgements

Chris T. Bauch and Madhur Anand are supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Madhur Anand also thanks the Canada Research Chairs program and The University of Guelph for support.

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