Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

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.     

The influence of generalist predators in spatially extended predator–prey systems

The presence of generalist predators is known to have important ecological impacts in several fields. They have wide applicability in the field of biological control. However, their role in the spatial distribution of predator and prey populations is still not clear. In this paper, the spatial dynamics of a predator–prey system is investigated by considering two different types of generalist predators. In one case, it is considered that the predator population has an additional food source and can survive in the absence of the prey population. In the other case, the predator population is involved in intraguild predation, i.e., the source of the additional food of the predator coincides with the food source of the prey population and thus both prey and predator populations compete for the same resource. The conditions for linear stability and Turing instability are analyzed for both the cases. In the presence of generalist predators, the system shows different pattern formations and spatiotemporal chaos which has important implications for ecosystem functioning not only in terms of their predictability, but also in influencing species persistence and ecosystem stability in response to abrupt environmental changes. This study establishes the importance of the consideration of spatial dynamics while determining optimal strategies for biological control through generalist predators.     

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.     

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.     

Individual base model of predator-prey system shows predator dominant dynamics

Individual base model of predator-prey system is constructed. Both predator and prey species have age structure and cohorts of early reproductive age have competitive advantage. The model has linear functional response in predation behavior and includes the effect of interference among predators and delay of population growth from resource intake, not by functional response but by calculation procedure. Each foraging action is calculated successively and surplus or scarce of acquired resources is interpreted into population size through individual birth and death. This model shows that biomass of prey killed by predator is dependent on demand of predator and that heterogeneity in predator population is essential in persistency and stability of predator-prey system. Heterogeneity of predator makes predator individuals of less competing ability die rapidly. Rapid death of weak individuals causes rapid decrease of total demand of predator and that makes enough room for survived predators. Therefore, the biomass of killed prey is dependent on predator's demand. As young or infant population of predator are the more vulnerable to shortage of prey, and when many of them cannot survive to reproductive age, they can stabilize the system by wasting excessive prey with only temporal numerical increase of predator population.     

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.     

Predator overcomes the Allee effect due to indirect prey–taxis

A mathematical model for spatiotemporal dynamics of prey–predator system was studied by means of linear analysis and numerical simulations. The model is a system of PDEs of taxis–diffusion–reaction type, accounting for the ability of predators to detect the locations of higher prey density, which is formalized as indirect prey–taxis, according to hypothesis that the taxis stimulus is a substance being continuously emitted by the prey, diffusing in space and decaying with constant rate in time (e.g. odour, pheromone, exometabolit). The local interactions of the prey and predators are described by the classical Rosenzweig – MacArthur system, which is modified in order to take into account the Allee effect in the predator population. The boundary conditions determine the absence of fluxes of population densities and stimulus concentration through the habitat boundaries. The obtained results suggest that the prey–taxis activity of the predator can destabilize both the stationary and periodic spatially-homogeneous regimes of the species coexistence, causing emergence of various heterogeneous patterns. In particular, it is demonstrated that formation of local dense aggregations induced by prey–taxis allows the predators to overcome the Allee effect in its population growth, avoiding the extinction that occurs in the model in the absence of spatial effects.     

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.     

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.     

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect

Invasion of an exotic species initiated by its local introduction is considered subject to predator-prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion-reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.     

Influence of consumer mutual interference on the stabilization of a three-level trophic system

In this paper, we present a three-level (food–prey–predator) trophic food chain which includes consumer mutual interference (MIF). In contrast with other analyses, we consider the effect of both prey and predator MIF on the dynamics of a three-level trophic system. MIF is generally considered to exert a stabilizing effect on population dynamics based on the predator–prey model. However, results from analytical and numerical simulations utilizing a simple three-species food chain model suggest that while the addition of prey MIF to the model provides a stabilizing influence, as the chaotic dynamics collapse to a stable steady state, adding only predator MIF to the model can only stabilize the system at intermediate MIF values. The three-species trophic food chain is also stabilized when combination of both prey and predator MIF is added to the model. Our work serves to provide insight into the effects of MIF in the real world.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

PREY ADAPTATION AS A CAUSE OF PREDATOR-PREY CYCLES

We analyze simple models of predator-prey systems in which there is adaptive change in a trait of the prey that determines the rate at which it is captured by searching predators. Two models of adaptive change are explored: (1) change within a single reproducing prey population that has genetic variation for vulnerability to capture by the predator; and (2) direct competition between two independently reproducing prey populations that differ in their vulnerability. When an individual predator's consumption increases at a decreasing rate with prey availability, prey adaptation via either of these mechanisms may produce sustained cycles in both species' population densities and in the prey's mean trait value. Sufficiently rapid adaptive change (e.g., behavioral adaptation or evolution of traits with a large additive genetic variance), or sufficiently low predator birth and death rates will produce sustained cycles or chaos, even when the predator-prey dynamics with fixed prey capture rates would have been stable. Adaptive dynamics can also stabilize a system that would exhibit limit cycles if traits were fixed at their equilibrium values. When evolution fails to stabilize inherently unstable population interactions, selection decreases the prey's escape ability, which further destabilizes population dynamics. When the predator has a linear functional response, evolution of prey vulnerability always promotes stability. The relevance of these results to observed predator-prey cycles is discussed.     

A predator-prey model with infected prey

A predator-prey model with logistic growth in the prey is modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds are identified which determine when the predator population survives and when the disease remains endemic. For some parameter values the greater vulnerability of the infected prey allows the predator population to persist, when it would otherwise become extinct. Also the predation on the more vulnerable prey can cause the disease to die out, when it would remain endemic without the predators.     

Disease in prey population and body size of intermediate predator reduce the prevalence of chaos-conclusion drawn from Hastings–Powell model

In ecology the disease in the prey population plays an important role in controlling the dynamical behaviour of the system. We modify Hastings and Powell’s (HP) [Hastings, A., Powell, T., 1991. Chaos in three-species food chain. Ecology 72 (3), 896–903] model by introducing disease in the prey population. The conditions for which the modified HP model system represents extinction, permanence or impermanence of population are worked out. The modified model is analyzed to obtain different conditions for which the system exhibits stability around the biologically feasible equilibria. Through numerical simulations we display that the modified system enters into stable solutions depending upon the force of infection in prey population as well as body size of intermediate predator. Our results demonstrate that disease in prey population and body size of intermediate predator are the key parameters for controlling the chaotic dynamics observed in original HP model.     

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.     

Should "handled" prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory

Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered.     

Periodic travelling waves in cyclic predator–prey systems

Predation is an established cause of cycling in prey species. Here, the ability of predation to explain periodic travelling waves in prey populations, which have recently been found in a number of spatiotemporal field studies, is examined. The nature of periodic waves in these systems, and the way in which they can be generated by the invasion of predators into a prey population is discussed. A theoretical calculation that predicts, as a function of two parameter ratios, whether such an invasion will lead to a stable periodic travelling wave that would be observed in practice is presented ‐ the alternative outcome is spatiotemporal chaos. The calculation also predicts quantitative details of the periodic waves, such as speed and amplitude. The results give new insights into the types of predator‐prey systems in which one would expect to see periodic travelling waves following an invasion by predators.     

Effect of disease-selective predation on prey infected by contact and external sources

We propose and analyze a simple mathematical model for susceptible prey (S)–infected prey (I)–predator (P) interaction, where the susceptible prey population (S) is infected directly from external sources as well as through contact with infected class (I) and the predator completely avoids consuming the infected prey. The model is analyzed to obtain different thresholds of the key parameters under which the system exhibits stability around the biologically feasible equilibria. Through numerical simulations we display the effects of external infection and the infection through contact on the system dynamics in the absence as well as in the presence of the predator. We compare the system dynamics when infection occurs only through contact, with that when it occurs through contact and external sources. Our analysis demonstrates that under a disease-selective predation, stability and oscillations of the system is determined by two key parameters: the external infection rate and the force of infection through contact. Due to the introduction of external infection, the predator and the prey population show limit-cycle oscillations over a range parametric values. We suggest that while predicting the dynamics of such an eco-epidemiological system, the modes of infection and the infection rates might be carefully investigated.