Community-driven dispersal in an individual-based predator-prey model

We present a spatial, individual-based predator-prey model in which dispersal is dependent on the local community. We determine species suitability to the biotic conditions of their local environment through a time and space varying fitness measure. Dispersal of individuals to nearby communities occurs whenever their fitness falls below a predefined tolerance threshold. The spatiotemporal dynamics of the model is described in terms of this threshold. We compare this dynamics with the one obtained through density-independent dispersal and find marked differences. In the community-driven scenario, the spatial correlations in the population density do not vary in a linear fashion as we increase the tolerance threshold. Instead we find the system to cross different dynamical regimes as the threshold is raised. Spatial patterns evolve from disordered, to scale-free complex patterns, to finally becoming well-organized domains. This model therefore predicts that natural populations, the dispersal strategies of which are likely to be influenced by their local environment, might be subject to complex spatiotemporal dynamics.Comment: 43 pages, 7 figures, vocabulary modifications, discussion expanded, references added, Ecological Complexity accepte

Bistability induced by generalist natural enemies can reverse pest invasions

Reaction-diffusion analytical modeling of predator-prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. Few additional analytical results have been obtained for other spatially distributed predator-prey systems, as traveling waves of non-monotonous systems are notoriously difficult to obtain. Traveling waves have indeed recently been shown to exist in predator-prey systems, but the direction of the wave, an essential item of information in the context of the control of biological invasions, is generally unknown. Preliminary numerical explorations have hinted that control by generalist predators might be possible for prey populations displaying logistic growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction-diffusion equations. The population of the focal prey - the invader - is assumed to grow according to a logistic function. The predator has a type II functional response and is present everywhere in the domain, at its carrying capacity, on alternative hosts. Control, defined as the invader becoming extinct in the domain, may result from spatially independent demographic dynamics or from a spatial extinction wave. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations (PDEs). The searching efficiency and functional response plateau of the predator are identified as the main parameters defining the parameter space for prey extinction and invasion. Numerical explorations are carried out in the region of those control parameters space between the super-and subso-lutions, in which no conclusion about controllability can be drawn on the basis of analytical solutions. The ability of generalist predators to control prey populations with logistic growth lies in the bis-table dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. The consideration of space in predator-prey systems involving generalist predators with a parabolic functional response is crucial. Analysis of the ordinary differential equations (ODEs) system identifies parameter regions with monostable (extinction) and bistable (extinction or invasion) dynamics. By contrast, analysis of the associated PDE system distinguishes different and additional regions of invasion and extinction. Depending on the relative positions of these different zones, four patterns of spatial dynamics can be identified : traveling waves of extinction and invasion, pulse waves of extinction and heterogeneous stationary positive solutions of the Turing type. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator-prey systems with specialist natural enemies

Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge

In real-world ecosystem, studies on the mechanisms of spatiotemporal pattern formation in a system of interacting populations deserve special attention for its own importance in contemporary theoretical ecology. The present investigation deals with the spatial dynamical system of a two-dimensional continuous diffusive predator–prey model involving the influence of intra-species competition among predators with the incorporation of a constant proportion of prey refuge. The linear stability analysis has been carried out and the appropriate condition of Turing instability around the unique positive interior equilibrium point of the present model system has been determined. Furthermore, the existence of the various spatial patterns through diffusion-driven instability and the Turing space in the spatial domain have been explored thoroughly. The results of numerical simulations reveal the dynamics of population density variation in the formation of isolated groups, following spotted or stripe-like patterns or coexistence of both the patterns. The results of the present investigation also point out that the prey refuge does have significant influence on the pattern formation of the interacting populations of the model under consideration

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigen modes acts as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response

Nonexistence of Periodic Orbits for Predator-Prey System with Strong Allee Effect in Prey Populations

We use Dulac criterion to prove the nonexistence of periodic orbits for a class of general predator-prey system with strong Allee effect in the prey population growth. This completes the global bifurcation analysis of typical predator-prey systems with strong Allee effect for all possible parameters

Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response

In this paper, we extend a Leslie–Gower-type predator–prey system with ratio-dependent Holling III functional response considering the cost of antipredator defence due to fear. We study the impact of the fear effect on the model, and we find that many interesting dynamical properties of the model can occur when the fear effect is present. Firstly, the relationship between the fear coefficient K and the positive equilibrium point is introduced. Meanwhile, the existence of the Turing instability, the Hopf bifurcation, and the Turing–Hopf bifurcation are analyzed by some key bifurcation parameters. Next, a normal form for the Turing–Hopf bifurcation is calculated. Finally, numerical simulations are carried out to corroborate our theoretical results

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single

Ecological system with fear induced group defence and prey refuge

In this study, we investigate the dynamics of a spatial and non spatial prey-predator interaction model that includes the following: (i) fear effect incorporated in prey birth rate; (ii) group defence of prey against predators; and (iii) prey refuge. We provide comprehensive mathematical analysis of extinction and persistence scenarios for both prey and predator species. To better explore the dynamics of the system, a thorough investigation of bifurcation analysis has been performed using fear level, prey birth rate, and prey death rate caused by intra-prey competition as bifurcation parameter. All potential occurrences of bi-stability dynamics have also been investigated for some relevant sets of parametric values. Our numerical evaluations show that high levels of fear can stabilize the prey-predator system by ruling out the possibility of periodic solutions. Also, our model Hopf bifurcation is subcritical in contrast to traditional prey-predator models, which ignore the cost of fear and have supercritical Hopf bifurcations in general. In contrast to the general trend, predator species go extinct at higher values of prey birth rates. We have also found that, contrary to the typical tendency for prey species to go extinct, both prey and predator populations may coexist in the system as intra-prey competition level grows noticeably. The stability and Turing instability of associated spatial model have also been investigated analytically. We also perform the numerical simulation to observe the effect of different parameters on the density distribution of species. Different types of spatiotemporal patterns like spot, mixture of spots and stripes have been observed via variation of time evolution, diffusion coefficient of predator population, level of fear factor and prey refuge. The fear level parameter (k) has a great impact on the spatial dynamics of model system