Revealing the role of predator interference in a predator–prey system with disease in prey population

Predation on a species subjected to an infectious disease can affect both the infection level and the population dynamics. There is an ongoing debate about the act of managing disease in natural populations through predation. Recent theoretical and empirical evidence shows that predation on infected populations can have both positive and negative influences on disease in prey populations. Here, we present a predator–prey system where the prey population is subjected to an infectious disease to explore the impact of predator on disease dynamics. Specifically, we investigate how the interference among predators affects the dynamics and structure of the predator–prey community. We perform a detailed numerical bifurcation analysis and find an unusually large variety of complex dynamics, such as, bistability, torus and chaos, in the presence of predators. We show that, depending on the strength of interference among predators, predators enhance or control disease outbreaks and population persistence. Moreover, the presence of multistable regimes makes the system very sensitive to perturbations and facilitates a number of regime shifts. Since, the habitat structure and the choice of predators deeply influence the interference among predators, thus before applying predators to control disease in prey populations or applying predator control strategy for wildlife management, it is essential to carefully investigate how these predators interact with each other in that specific habitat; otherwise it may lead to ecological disaster.     

Cooperative hunting in a discrete predator–prey system II

ABSTRACT

We investigate a discrete-time predator–prey system with cooperative hunting in the predators proposed by Chow et al. by determining local stability of the interior steady states analytically in certain parameter regimes. The system can have either zero, one or two interior steady states. We provide criteria for the stability of interior steady states when the system has either one or two interior steady states. Numerical examples are presented to confirm our analytical findings. It is concluded that cooperative hunting of the predators can promote predator persistence but may also drive the predator to a sudden extinction.     

Asexual reproduction changes predator population dynamics in a life predator–prey system

Many organisms display oscillations in population size. Theory predicts that these fluctuations can be generated by predator–prey interactions, and empirical studies using life model systems, such as a rotifer-algae community consisting of Brachionus calyciflorus as predator and Chlorella vulgaris as prey, have been successfully used for studying such dynamics. B. calyciflorus is a cyclical parthenogen (CP) and clones often differ in their sexual propensity, that is, the degree to which they engage into sexual or asexual (clonal) reproduction. Since sexual propensities can affect growth rates and population sizes, we hypothesized that this might also affect population oscillations. Here, we studied the dynamical behaviour of B. calyciflorus clones representing either CPs (regularly inducing sex) or obligate parthenogens (OPs). We found that the amplitudes of population cycles to be increased in OPs at low nutrient levels. Several other population dynamic parameters seemed unaffected. This suggests that reproductive mode might be an important additional variable to be considered in future studies of population oscillations.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Evolution towards oscillation or stability in a predator–prey system

We studied a prey–predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey''s (host''s) defence is effective when compared with the predator''s (parasite''s) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator''s (parasite''s) attacking ability exceeds the prey''s (host''s) defensive ability. Second, a stable equilibrium is more likely if the prey''s defensive trait evolves faster than the predator''s attack trait, whereas population oscillations are likely if the predator''s trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey''s abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host–parasite systems are more likely than those in prey–predator systems to show large-amplitude oscillations.     

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.     

On the Neimark–Sacker bifurcation in a discrete predator–prey system

A two-parameter family of discrete models describing a predator–prey interaction is considered, which generalizes a model discussed by Murray, and originally due to Nicholson and Bailey, consisting of two coupled nonlinear difference equations. In contrast to the original case treated by Murray, where the two populations either die out or may display unbounded growth, the general member of this family displays a somewhat wider range of behaviour. In particular, the model has a nontrivial steady state which is stable for a certain range of parameter values, which is explicitly determined, and also undergoes a Neimark–Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter space and a repelling one in others.     

Water turbidity affects predator–prey interactions in a fish–damselfly system

Community structure may differ dramatically between clear-water and turbid lakes. These differences have been attributed to differences in the cascading effect of fish on prey populations, owing to the reduced efficiency of fish predation in the presence of macrophytes. However, recent theoretical ideas suggest that water turbidity may shape predator–prey interactions, and it is predicted that prey will relax its antipredation behaviour in turbid water (H1). As a result, the nature of predator–prey interactions is expected to shift from both direct and indirect in clear water to dominantly direct in turbid water (H2). We tested these ideas in a fish–damselfly predator–prey system. In a first behavioural experiment, we looked at antipredation behaviour of damselfly larvae isolated from habitats that differ in turbidity, in the presence of fish in clear and turbid water. As predicted in H1, the larvae were more active in turbid than in clear water. In a complementary enclosure experiment, we reared larvae in a clear-water pond and a turbid pond, respectively, and manipulated the origin of the larvae (clear-water, turbid pond), fish presence (absent, present), and vegetation density (sparse, abundant). In both ponds, fish had a direct negative effect on survival of the larvae, which was mitigated in the presence of vegetation. In the fish treatment, the change in average body mass tended to be higher in the turbid pond than in the clear-water pond, suggesting indirect effects of fish were mitigated in the turbid pond. This was supported by a negative effect of fish on the effective growth rate of larvae in the clear pond, but not in the turbid pond. These results are compatible with the idea that predator–prey relationships are mainly governed by direct effects in turbid water, and by direct and indirect effects in clear water.     

On a predator–prey system of Gause type

In this paper a Gause type model of interactions between predator and prey population is considered. We deal with the sufficient condition due to Kuang and Freedman in the generalized form including a kind of weight function. In a previous paper we proved that the existence of such weight function implies the uniqueness of limit cycle. In the present paper we give a new condition equivalent to the existence of a weight function (Theorem 4.4). As a consequence of our result, it is shown that some simple qualitative properties of the trophic function and the prey isocline ensure the uniqueness of limit cycle.     

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.     

Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

Spatiotemporal behavior of a prey–predator system with a group defense for prey

Group defense is a strategy widely employed by various species. We consider the effect of grouping on population persistence when animals join together in herds in order to provide a self-defense from predators. In literature, group defense is usually addressed in terms of individual behavioral responses. In this paper, we consider an alternative ‘mean-field’ approach which uses prey and predator densities as the dynamical variables. The model is essentially a predator–prey system but with an unconventional parametrization for the predation term. We discuss the outcomes of the ecosystem dynamics in terms of persistence and prey survival. In the spatially distributed model some specific spatio-temporal features are discovered.     

Effects of perturbation on the predator–prey system in a heterogeneous landscape

Environmental perturbations occur in ecosystems as the result of disturbance, which is closely related to ecosystem stability and resilience. To understand how perturbations can affect ecosystems, we constructed a spatially explicit lattice model to simulate the integrative predator–prey–grass relationships. In this model, a predator (or prey) gives birth to offspring, according to a specific birth probability, when it is able to feed on prey (or grass). When a predator or prey animal was initially introduced or newly born, its health state was set at a given high value. This state decreased by 1 with each time step. When the state of an animal decreased to zero, the animal was considered dead and was removed from the system. In this model, the perturbation was defined as the sudden death of some portion of the population. The heterogeneous landscape was characterized by a parameter, H, which controlled the degree of heterogeneity. When H ≥ 0.6, the predator population size was positively influenced by the perturbation. However, the perturbation had little effect upon the population sizes of prey or grass, regardless of the value of H.     

Environmental fluctuations restrict eco-evolutionary dynamics in predator–prey system

Environmental fluctuations, species interactions and rapid evolution are all predicted to affect community structure and their temporal dynamics. Although the effects of the abiotic environment and prey evolution on ecological community dynamics have been studied separately, these factors can also have interactive effects. Here we used bacteria–ciliate microcosm experiments to test for eco-evolutionary dynamics in fluctuating environments. Specifically, we followed population dynamics and a prey defence trait over time when populations were exposed to regular changes of bottom-up or top-down stressors, or combinations of these. We found that the rate of evolution of a defence trait was significantly lower in fluctuating compared with stable environments, and that the defence trait evolved to lower levels when two environmental stressors changed recurrently. The latter suggests that top-down and bottom-up changes can have additive effects constraining evolutionary response within populations. The differences in evolutionary trajectories are explained by fluctuations in population sizes of the prey and the predator, which continuously alter the supply of mutations in the prey and strength of selection through predation. Thus, it may be necessary to adopt an eco-evolutionary perspective on studies concerning the evolution of traits mediating species interactions.     

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.     

Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey, the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain. For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding of population dynamics on a fragmented habitat.     

Prediction of predator–prey populations modelled by perturbed ODEs

In this paper we explore a stochastic model in continuous time for predator-prey interactions, which accounts for the periodical behaviour observed in many animal populations. More precisely, we consider a solution to the classical Lotka-Volterra system of equations, but we view the actual population sizes as random perturbations of the solutions to this ODE system. Namely, we assume that the perturbations follow correlated Ornstein-Uhlenbeck processes; this approach generalizes the one of Froda and Colavita [Aust N Z J Stat 2:235-254, 2005] who considered only i.i.d. errors. This type of perturbed deterministic model allows to perform parameter estimation and to predict population sizes at future times. On the other hand, the present model refines the previous one since it takes into account the variability due to external factors and the time dependence in the random component. Moreover, this more flexible model improves the predictions of population sizes at future times. In order to illustrate this last point, we analyse two data sets.     

Turing patterns in a predator–prey model with seasonality

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction–diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the ‘summer’ season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington–DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).

     

A predator–prey refuge system: Evolutionary stability in ecological systems

A refuge model is developed for a single predator species and either one or two prey species where no predators are present in the prey refuge. An individual’s fitness depends on its strategy choice or ecotype (predators decide which prey species to pursue and prey decide what proportion of their time to spend in the refuge) as well as on the population sizes of all three species. It is shown that, when there is a single prey species with a refuge or two prey species with no refuge compete only indirectly (i.e. there is only apparent competition between prey species), that stable resident systems where all individuals in each species have the same ecotype cannot be destabilized by the introduction of mutant ecotypes that are initially selectively neutral. In game-theoretic terms, this means that stable monomorphic resident systems, with ecotypes given by a Nash equilibrium, are both ecologically and evolutionarily stable. However, we show that this is no longer the case when the two indirectly-competing prey species have a refuge. This illustrates theoretically that two ecological factors, that are separately stabilizing (apparent competition and refuge use), may have a combined destabilizing effect from the evolutionary perspective. These results generalize the concept of an evolutionarily stable strategy (ESS) to models in evolutionary ecology. Several biological examples of predator–prey systems are discussed from this perspective.