Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator–prey model

Many natural systems are subject to seasonal environmental change. As a consequence many species exhibit seasonal changes in their life history parameters—such as a peak in the birth rate in spring. It is important to understand how this seasonal forcing affects the population dynamics. The main way in which seasonal models have been studied is through a two dimensional bifurcation approach. We augment this bifurcation approach with extensive simulation in order to understand the potential solution behaviours for a predator–prey system with a seasonally forced prey growth rate. We consider separately how forcing influences the system when the unforced dynamics have either monotonic decay to the coexistence steady state, or oscillatory decay, or stable limit cycles. The range of behaviour the system can exhibit includes multi-year cycles of different periodicities, parameter ranges with coexisting multi-year cycles of the same or different period as well as quasi-periodicity and chaos. We show that the level of oscillation in the unforced system has a large effect on the range of behaviour when the system is seasonally forced. We discuss how the methods could be extended to understand the dynamics of a wide range of ecological and epidemiological systems that are subject to seasonal changes.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.     

A nonlinear,second-order population model

Delay differential, difference, and partial differential equation models are being used more extensively to explain single-species population oscillations and limit cycle behavior. Ordinary differential equation (ODE) models have been largely ignored. This is because first-order ODE models are inherently monotonic. Certainly this is not usual population behavior in the real world. If it is assumed that the per capita growth rate of a population changes over time as a result of regulating factors impinging on it, then a more realistic biological model results. The model translates into a second-order nonlinear ODE. Such a model can exhibit oscillatory and limit cycle as well as monotonic solutions, i.e., behavior for which non-ODE models have been used to explain. Although first-order ODE models are gross simplifications of real phenomena, ODE models in general should not be disregarded as important analytical tools.     

Effect of diffusion and spatially varying predation risk on the dynamics and equilibrium density of a predator-prey system

Starting from natural planktonic systems, we present a new mechanism involving spatial heterogeneity, and develop a new spatial structure model of planktonic predation systems. Firstly, the effect of diffusion on the dynamics of the system is investigated. We find that diffusion of only prey or both prey and predator between different patches with different predation risk may stabilize the dynamics, depending on the flow rate. Only a medium flow rate can lead to the stability of the system. Too large a rate can cause the system to approach the non-spatial limit case of a well-mixed system. Too large a rate can cause the system to approach the non-spatial limit case as a well-mixed system, which is characterized by its strongly oscillatory dynamics. When only prey diffuse, the smaller the parameter f (the proportion of the patchy volume with larger predation risk to the total volume), the more stable the system. If both populations can diffuse, however, only medium and very small f values may stabilize the system. Also, the response of the spatially averaged equilibrium densities of the system to the increasing of the flow rate is examined. With increasing flow rate, the spatial-averaged equilibrium density of prey decreases, while that of predator depends on which species can diffuse. For the case of prey diffusion only, it first remains unchanged and then slightly decreases, while it increases for the case of combinations as the flow rate increases. Our results are, qualitatively, determined by the spatially heterogeneous mechanism that we propose, and further regulated by top-down forces. Of practical importance, the results reported here indicate that which species diffuse plays a key role in the ways in which diffusion influences the dynamics and the spatial-average equilibrium densities of the system responses to the flow rate's increasing.     

From pattern to process: identifying predator–prey models from time-series data

Fitting nonlinear models to time-series is a technique of increasing importance in population ecology. In this article, we apply it to assess the importance of predator dependence in the predation process by comparing two alternative models of equal complexity (one with and one without predator dependence) to predator–prey time-series. Stochasticities in such data come from both observation error and process error. We consider how these errors must be taken into account in the fitting process, and we develop eight different model selection criteria. Applying these criteria to laboratory data on simple protozoan and arthropod predator–prey systems shows that little predator dependence is present, with one interesting exception. Field data are more ambiguous (either selection depends on the particular criterion or no significant differences can be detected), and we show that both models fit reasonably well. We conclude that, within our modeling framework, predator dependence is in general insignificant in simple systems in homogeneous environments. Relatively complex systems show significant predator dependence more often than simple ones but the data are also often inconclusive. The analysis of such systems should rely on several models to detect predictions that are sensitive to predator dependence and to direct further research if necessary. Received: July 13, 2000 / Accepted: September 25, 2001     

Dynamics and responses to mortality rates of competing predators undergoing predator-prey cycles

Two or more competing predators can coexist using a single homogeneous prey species if the system containing all three undergoes internally generated fluctuations in density. However, the dynamics of species that coexist via this mechanism have not been extensively explored. Here, we examine both the nature of the dynamics and the responses of the mean densities of each predator to mortality imposed upon it or its competitor. The analysis of dynamics uncovers several previously undescribed behaviors for this model, including chaotic fluctuations, and long-term transients that differ significantly from the ultimate patterns of fluctuations. The limiting dynamics of the system can be loosely classified as synchronous cycles, asynchronous cycles, and chaotic dynamics. Synchronous cycles are simple limit cycles with highly positively correlated densities of the two predator species. Asynchronous cycles are limit cycles, frequently of complex form, including a significant period during which prey density is nearly constant while one predator gradually, monotonically replaces the other. Chaotic dynamics are aperiodic and generally have intermediate correlations between predator densities. Continuous changes in density-independent mortality rates often lead to abrupt transitions in mean population sizes, and increases in the mortality rate of one predator may decrease the population size of the competing predator. Similarly, increases in the immigration rate of one predator may decrease its own density and increase the density of the other predator. Proportional changes in one predator's birth and death rate functions can have significant effects on the dynamics and mean densities of both predator species. All of these responses to environmental change differ from those observed when competitors coexist stably as the result of resource (prey) partitioning. The patterns described here occur in many other competition models in which there are cycles and differences in the linearity of the responses of consumers to their resources.     

Compensation and stability in nonlinear matrix models.

Stability, bifurcation, and dynamic behavior, investigated here in discrete, nonlinear, age-structured models, can be complex; however, restrictions imposed by compensatory mechanisms can limit the behavioral spectrum of a dynamic system. These limitations in transitional behavior of compensatory models are a focal point of this article. Although there is a tendency for compensatory models to be stable, we demonstrate that stability in compensatory systems does not always occur; for example, equilibria arising through a bifurcation can be initially unstable. Results concerning existence and uniqueness of equilibria, stability of the equilibria, and boundedness of solutions suggest that "compensatory" systems might not be compensatory in the literal sense.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003     

Complex dynamics of a predator–prey–parasite system: An interplay among infection rate,predator's reproductive gain and preference

Parasites are considered as an important factor in regulating their host populations through trait-mediated effects. On the other hand, predation becomes particularly interesting in host–parasite systems because predation can significantly alter the abundance of parasites and their host population. The combined effects of parasites and predator on host population and community structure therefore may have larger effect. Different field experiments confirm that predators consume disproportionately large number of infected prey in comparison to their susceptible counterpart. There are also substantial evidences that predator has the ability to distinguish prey that have been infected by a parasite and avoid such prey to reduce fitness cost. In this paper we study the predator–prey dynamics, where the prey species is infected by some parasites and predators consume both the susceptible and infected prey with some preference. We demonstrate that complexity in such systems largely depends on the predator's selectivity, force of infection and predator's reproductive gain. If the force of infection and predator's reproductive gain are low, parasites and predators both go to extinction whatever be the predator's preference. The story may be totally different in the opposite case. Survival of species in stable, oscillatory or chaotic states, and their extinction largely depend on the predator's preference. The system may also show two coexistence equilibrium points for some parameter values. The equilibrium with lower susceptible prey density is always stable and the equilibrium with higher susceptible prey density is always unstable. These results suggest that understanding the consequences of predator's selectivity or preference may be crucial for community structure involving parasites.     

Complexity in a prey-predator model with prey refuge and diffusion

Prey-predator interaction is one of the most commonly observed relationships in ecosystem. In the study of prey-predator models, it is frequently assumed that the changes in population densities are only time-dependent and the dynamics is generally represented by coupled nonlinear ordinary differential equations. In natural system, however, either prey or predator or both move from one place to another for various reasons. In such a case, their dynamic interaction depends both on time and space and requires coupled nonlinear partial differential equations for its dynamic representation. It is also well documented that prey refuges affect the interaction between prey and predator significantly. In this paper, we studied the dynamics of a diffusive prey-predator interaction with prey refuge and type III response function. We have considered both one and two dimensional diffusivity in the model system and presented different stability results under the assumptions that one or both species may be mobile or sedentary. Our results showed that the system may exhibit different spatiotemporal (non-Turing) patterns, like spiral waves, patchy structures, spot pattern, or even spatiotemporal chaos depending on the refuge availability and diffusion rate of species. Another interesting finding was that the dynamic complexity in a prey-predator model increases in case of mobile predator and sedentary prey compare to mobile prey and sedentary predator while refuge availability is varied.     

Conditions for periodic solutions of volterra differential systems

As a contribution to the discussion of oscillatory models for interacting species it is shown that two-species Volterra models can never have limit cycles, and a complete enumeration is given of conditions which the parameters of these models must satisfy in order that a part of the phase space be filled with a family of closed curves; sketches of phase portraits are also given. These results complement and correct older results by Bautin and by Coppel on quadratic differential systems. The paper opens with a brief discussion of some more practical aspects of the ecological application of oscillatory models.     

The influence of vigilance on intraguild predation

Theoretical work on intraguild predation suggests that if a top predator and an intermediate predator share prey, the system will be stable only if the intermediate predator is better at exploiting the prey, and the top predator gains significantly from consuming the intermediate predator. In mammalian carnivore systems, however, there are examples of top predator species that attack intermediate predator species, but rarely or never consume the intermediate predator. We suggest that top predators attacking intermediate predators without consuming them may not only reduce competition with the intermediate predators, but may also increase the vigilance of the intermediate predators or alter the vigilance of their shared prey, and that this behavioral response may help to maintain the stability of the system. We examine two models of intraguild predation, one that incorporates prey vigilance, and a second that incorporates intermediate predator vigilance. We find that stable coexistence can occur when the top predator has a very low consumption rate on the intermediate predator, as long as the attack rate on the intermediate predator is relatively large. However, the system is stable when the top predator never consumes the intermediate predator only if the two predators share more than one prey species. If the predators do share two prey species, and those prey are vigilant, increasing top predator attack rates on the intermediate predator reduces competition with the intermediate predator and reduces vigilance by the prey, thereby leading to higher top predator densities. These results suggest that predator and prey behavior may play an important dynamical role in systems with intraguild predation.     

Bistability and oscillations in the Huang-Ferrell model of MAPK signaling

Physicochemical models of signaling pathways are characterized by high levels of structural and parametric uncertainty, reflecting both incomplete knowledge about signal transduction and the intrinsic variability of cellular processes. As a result, these models try to predict the dynamics of systems with tens or even hundreds of free parameters. At this level of uncertainty, model analysis should emphasize statistics of systems-level properties, rather than the detailed structure of solutions or boundaries separating different dynamic regimes. Based on the combination of random parameter search and continuation algorithms, we developed a methodology for the statistical analysis of mechanistic signaling models. In applying it to the well-studied MAPK cascade model, we discovered a large region of oscillations and explained their emergence from single-stage bistability. The surprising abundance of strongly nonlinear (oscillatory and bistable) input/output maps revealed by our analysis may be one of the reasons why the MAPK cascade in vivo is embedded in more complex regulatory structures. We argue that this type of analysis should accompany nonlinear multiparameter studies of stationary as well as transient features in network dynamics.     

The effect of predator limitation on the dynamics of simple food chains

We investigate the influence of competition between predators on the dynamics of bitrophic predator–prey systems and of tritrophic food chains. Competition between predators is implemented either as interference competition, or as a density-dependent mortality rate. With interference competition, the paradox of enrichment is reduced or completely suppressed, but otherwise, the dynamical behavior of the systems is not fundamentally different from that of the Rosenzweig–MacArthur model, which contains no predator competition and shows only continuous transitions between fixed points or periodic oscillations. In contrast, with density-dependent predator mortality, the system shows a surprisingly rich dynamical behavior. In particular, decreasing the density regulation of the predator can induce catastrophic shifts from a stable fixed point to a large oscillation where the predator chases the prey through a cycle that brings both species close to the threshold of extinction. Other catastrophic bifurcations, such as subcritical Hopf bifurcations and saddle-node bifurcations of limit cycles, do also occur. In tritrophic food chains, we find again that fixed points in the model with predator interference become unstable only through Hopf bifurcations, which can also be subcritical, in contrast to the bitrophic situation. The model with a density limitation shows again catastrophic destabilization of fixed points and various nonlocal bifurcations. In addition, chaos occurs for both models in appropriate parameter ranges.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

The role of seasonality in the dynamics of deer tick populations

In this paper, we formulate a nonlinear system of difference equations that models the three-stage life cycle of the deer tick over four seasons. We study the effect of seasonality on the stability and oscillatory behavior of the tick population by comparing analytically the seasonal model with a non-seasonal one. The analysis of the models reveals the existence of two equilibrium points. We discuss the necessary and sufficient conditions for local asymptotic stability of the equilibria and analyze the boundedness and oscillatory behavior of the solutions. A main result of the mathematical analysis is that seasonality in the life cycle of the deer tick can have a positive effect, in the sense that it increases the stability of the system. It is also shown that for some combination of parameters within the stability region, perturbations will result in a return to the equilibrium through transient oscillations. The models are used to explore the biological consequences of parameter variations reflecting expected environmental changes.     

Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

We study the influence of nonlocal intraspecies prey competition on the spatiotemporal patterns arising behind predator invasions in two oscillatory reaction–diffusion integro-differential models. We use three common types of integral kernels as well as develop a caricature system, to describe the influence of the standard deviation and kurtosis of the kernel function on the patterns observed. We find that nonlocal competition can destabilize the spatially homogeneous state behind the invasion and lead to the formation of complex spatiotemporal patterns, including stationary spatially periodic patterns, wave trains and irregular spatiotemporal oscillations. In addition, the caricature system illustrates how large standard deviation and low kurtosis facilitate the formation of these spatiotemporal patterns. This suggests that nonlocal competition may be an important mechanism underlying spatial pattern formation, particularly in systems where the competition between individuals varies over space in a platykurtic manner.     

Complex Dynamics in an Eco-epidemiological Model

The presence of infectious diseases can dramatically change the dynamics of ecological systems. By studying an SI-type disease in the predator population of a Rosenzweig–MacArthur model, we find a wealth of complex dynamics that do not exist in the absence of the disease. Numerical solutions indicate the existence of saddle–node and subcritical Hopf bifurcations, turning points and branching in periodic solutions, and a period-doubling cascade into chaos. This means that there are regions of bistability, in which the disease can have both a stabilising and destabilising effect. We also find tristability, which involves an endemic torus (or limit cycle), an endemic equilibrium and a disease-free limit cycle. The endemic torus seems to disappear via a homoclinic orbit. Notably, some of these dynamics occur when the basic reproduction number is less than one, and endemic situations would not be expected at all. The multistable regimes render the eco-epidemic system very sensitive to perturbations and facilitate a number of regime shifts, some of which we find to be irreversible.     

Can environmental variation generate positive indirect effects in a model of shared predation?

Classic models of apparent competition predict negative indirect effects between prey with a shared enemy. If predator per capita growth rates are nonlinear, then endogenously generated periodic cycles are predicted to generate less negative or even positive indirect effects between prey. Here I determine how exogenous mechanisms such as environmental variation could modify indirect effects. I find that exogenous variation can have a broader range of effects on indirect interactions than endogenously generated cycles. Indirect effects are altered by environmental variation even in simple models for which the per capita growth rate of the predator species is a linear function of population densities. Temporal variation that affects the predator attack rate or the conversion efficiency can lead to large increases or decreases in the indirect effects between prey, dependent on how prey populations co-vary with the environmental variation. Positive indirect effects can occur when the period of environmental variation is close to the natural period of the biological system and shifts in subharmonic resonance occur with the addition of the second prey. Models that include nonlinear numerical responses generally lead to indirect effects that are sensitive to environmental variation in more parameters and across a wider range of frequencies.