Nonconstant Positive Steady States and Pattern Formation of 1D Prey-Taxis Systems

Prey-taxis is the process that predators move preferentially toward patches with highest density of prey. It is well known to have an important role in biological control and the maintenance of biodiversity. To model the coexistence and spatial distributions of predator and prey species, this paper concerns nonconstant positive steady states of a wide class of prey-taxis systems with general functional responses over 1D domain. Linearized stability of the positive equilibrium is analyzed to show that prey-taxis destabilizes prey–predator homogeneity when prey repulsion (e.g., due to volume-filling effect in predator species or group defense in prey species) is present, and prey-taxis stabilizes the homogeneity otherwise. Then, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis. Moreover, we provide detailed and thorough calculations to determine properties such as pitchfork and turning direction of the local branches. Our stability results also provide a stable wave mode selection mechanism for thee reaction–advection–diffusion systems including prey-taxis models considered in this paper. Finally, we provide numerical studies of prey-taxis systems with Holling–Tanner kinetics to illustrate and support our theoretical findings. Our numerical simulations demonstrate that the \(2\times 2\) prey-taxis system is able to model the formation and evolution of various striking patterns, such as spikes, periodic oscillations, and coarsening even when the domain is one-dimensional. These dynamics can model the coexistence and spatial distributions of interacting prey and predator species. We also give some insights on how system parameters influence pattern formation in these models.     

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.     

Investigating the Turing Conditions for Diffusion-driven Instability in Predator-prey System with Hunting Cooperation Functional Response

In this paper, we focus on stability analysis of steady-state solutions of a predator-prey system with hunting cooperation functional response. The results show that the Turing instability can be affected not only the existence of hunting cooperation, but also the diffusion coefficients: (1) in the absence of predator diffusion, diffusion-driven instability can be induced by hunting cooperation, but no stable patterns appear; (2) the system can occur diffusion-driven instability and Turing patterns, when both predator and prey have diffusion, and the diffusion coefficient of prey is greater than that of the predator. The numerical simulations of two cases are presented to verify the validity of our theoretical results.     

Effect of prey-taxis on predator’s invasion in a spatially heterogeneous environment

In this study, we consider a diffusive predator–prey system with prey-taxis and ratio-dependent functional responses in a spatially heterogeneous environment. Prey-taxis implies that the predator exhibits directed movement in the presence of prey. We claim that when the predator diffuses uniformly without prey-taxis, only the diffusion of the species plays a role in the predator’s invasion in a spatially homogeneous environment. However, when the predator disperses through uniform diffusion together with prey-taxis, we observe that both the diffusion of species and prey-taxis affect the invasion of the predator, which is not the case in a spatially homogeneous environment. The results are obtained by investigating the local stability of a semi-trivial solution, using an eigenvalue analysis.     

Predator invasion in predator–prey model with prey-taxis in spatially heterogeneous environment

We consider a predator–prey model with prey-taxis and Holling-type II functional responses in a spatially heterogeneous environment to analyze the effects of prey-taxis and the heterogeneity of an environment on predator invasion. To achieve our goal, we investigate the stability of semi-trivial solution in which the predator is absent. It is known that both the predator diffusion and the death rate contribute to the predator invasion in a heterogeneous habitat when there is no prey-taxis. In this paper, we show that predator invasion is affected by the prey-taxis and diffusions of the prey-taxis model for a certain range of predator death rates in a heterogeneous environment. Furthermore, in cases where predator invasion by predator diffusion does not occur in a particular death rate range of the predator, predator invasion can occur by prey-taxis in a spatially heterogeneous habitat. In addition, we compare this phenomenon to the corresponding predator–prey model with ratio-dependent functional responses. It is observed that none of the predator’s diffusion and prey-taxis affect the predator’s invasion, and that only the predator’s death rate contributes to predator invasion for the model with ratio-dependent functional responses.     

Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition

This article discusses a predator–prey system with predator saturation and competition functional response. The local stability, existence of a Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained in the absence of diffusion. Further, we discuss the diffusion-driven instability, Hopf bifurcation for corresponding diffusion system with zero flux boundary condition and Turing instability region regarding the parameters are established. Finally, numerical simulations supporting the theoretical analysis are also included.     

Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion

Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.     

Positive steady-state solutions for predator–prey systems with prey-taxis and Dirichlet conditions

This paper is concerned with positive steady-state solutions of a class of cross-diffusion systems which arise in the study of the predator–prey systems with prey-taxis. Under homogeneous Dirichlet boundary conditions, we use the theory of fixed point index in positive cones to establish the existence of positive steady-state solutions. By analyzing two related eigenvalues, we further obtain the coexistence region with respect to the growth rates of two species and characterize the differences of coexistence region if different predator–prey interactions are adopted. Additionally, we investigate the limiting behavior of positive steady-state solutions as some parameter tends to infinity. Our results not only generalize the previously known one, but also present some new conclusions.     

Spatiotemporal dynamics of reaction–diffusion models of interacting populations

The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator–prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

比率依赖型捕食者-食饵系统行波解的存在性

本文讨论一类比率依赖型捕食者 -食饵系统的反应扩散方程组 .首先 ,我们证明了时间周期定常解的存在性和稳定性 .其次 ,我们给出了扩散引起正常数平衡解失稳的条件 .最后 ,我们证明了比率依赖型捕食者 -食饵系统行波解的存在性和渐近性 .     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

Spatial pattern formations in diffusive predator-prey systems with non-homogeneous Dirichlet boundary conditions

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.     

Dynamical complexity induced by Allee effect in a predator–prey model

In this paper, we investigate the complex dynamics induced by Allee effect in a predator–prey model. For the non-spatial model, Allee effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee effect, the model is bistable. For the spatial model, without Allee effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee effect is not simple, but rich and complex.     

The role of indirect prey-taxis and interference among predators in pattern formation

It is important to find biological factors that may lead to formation of patches in the distribution of species. We build a simple model describing a consumer/predator which, besides random dispersion, searches for food by moving toward the gradient of some chemical released by prey. This mechanism is referred as indirect prey-taxis. The predator's rate of consumption is assumed to drop due to interference among predators when too many of them encounter on some aggregate of prey. The latter is assumed to have a negligible motility with its density governed by an ODE. The interference among consumers is modeled by a modification of the Beddington's de Angelis functional response. Detailed bifurcation analysis and numerical simulations of an auxiliary system indicate that nonconstant steady states imitating patches of species occur in the model provided the predator density exceeds certain threshold and taxis strength is big enough.     

Spatiotemporal complexity of a discrete space-time predator−prey system with self- and cross-diffusion

In this research, we investigate the spatiotemporal dynamics of a discrete space-time predator−prey system with self- and cross-diffusion. Through stability analysis and bifurcation analysis, Turing pattern formation conditions are derived and two nonlinear mechanisms of pattern formation are found, i.e., pure Turing instability and Hopf-Turing instability. Numerical simulations reveal rich dynamics of the discrete predator−prey system. In spatially homogeneous case, stable homogeneous stationary states, homogeneous periodic, quasiperiodic and chaotic oscillating states are exhibited; in spatially heterogeneous case, a surprising variety of prey and predator patterns are described, including spotted, striped, labyrinth, gapped, spiral, circled patterns and many intermediate patterns. Moreover, sensitivity of spatiotemporal pattern formation to initial conditions is predicted along with Hopf-Turing instability, suggesting the self-organization of diverse patterns under identical parametric conditions. In comparison with former results in literature, the discrete version of reaction-diffusion model developed in this research capture more complicated and richer nonlinear dynamical behaviors, contributing to a new comprehending on the complex pattern formation of spatially extended discrete predator−prey systems.     

A reaction–diffusion system modeling predator–prey with prey-taxis

We are concerned with a system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator's velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation.     

Interactions of Turing and Hopf bifurcations in an additional food provided diffusive predator-prey model

Complex spatiotemporal dynamics of a diffusive predator-prey system involving additional food supply to predator and intra-specific competition among predator, are investigated. We establish critical conditions of the occurrence of Turing instability, which are necessary and sufficient. Furthermore, we also establish conditions of the occurrence of codimension-2 Turing-Hopf bifurcation and Turing-Turing bifurcation, by exploring interactions of Turing bifurcations and Hopf bifurcation. For Turing-Hopf bifurcation, by analyzing normal form truncated to order 3 which are derived by applying normal form method, it is shown that under proper conditions, diffusive predator-prey system generates interesting spatial, temporal and spatiotemporal patterns, including a pair of spatially inhomogeneous steady states, a spatially homogeneous periodic solution and a pair of spatially inhomogeneous periodic solutions. And numerical simulations are also shown to support theory analysis. Moreover, it is found that proper intra-specific competition among predator helps generate complex spatiotemporal dynamics. And, proper additional food supply to predator helps control the population fluctuations of predator and prey, while large quantity and high quality of additional food supply will lead to the extinction of prey and make predator change the source of food, which finally destroy the ecosystem. These investigations might help understand complex spatiotemporal dynamics of predator-prey system involving additional food supply to predator and intra-specific competition among predator, and help conserve species in an ecosystem via supplying suitable additional food.     

Stability and persistence of two-dimensional patterns

The canonical equations for evolution of the amplitude order parameters order parameters describing the nonlinear development and persistence of two-dimensional three-mode spatial patterns generated by Turing instability in dissipative systems are considered. The stability conditions for persistent hexagonal patterns are generalized, and the conditions under which patterns are either disrupted, exhibit bounded quasiperiodic or chaotic behavior, or decay under nonlinear evolution are derived. These conditions are applied to the specific three-mode amplitude evolution equations derived for the Schnakenberg model and a delay predator system in Chapter 3. Numerical results are presented for the persistence, disruption and decay of patterns in these systems, including fairly detailed comparisons with simulations results for the Snackenberg model.     

Mathematical study of two-variable systems with adaptive numerical methods

In this paper, we consider reaction–diffusion systems arising from two-component predator–prey models with Smith growth functional response. The mathematical approach used here is in two folds since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with the fourth-order difference operator and advance the resulting nonlinear system of ordinary differential equations with the two competing families of the exponential time differencing (ETD) schemes, and we analyze them for stability. Numerical comparison between these two methods for solving various predator–prey population models with functional responses are also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.