Spatial Dynamics of a Diffusive Prey-predator Model with Stage Structure and Fear Effect

To understand the influence of fear effect on population dynamics, especially for the populations with obvious stage structure characteristics, we propose and investigate a diffusive prey-predator model with stage structure in predators. First, we discuss the existence and stability of equilibrium of the model in the absence of diffusion. Then, we obtain the critical conditions for Hopf and Turing bifurcations. Some numerical simulations are also carried out to verify our theoretical results, which indicate that the fear can induce the prey population to show five pattern structures: cold-spot pattern, mixed pattern with cold spots and stripes, stripes pattern, hot-spot pattern, mixed pattern with hot spots and stripes. These findings imply that the fear effect induced by the mature predators plays an important role in the spatial distribution of species.     

Nonexistence of Nonconstant Positive Steady States of a Diffusive Predator-prey Model with Fear Effect

In this paper, we investigate a diffusive predator-prey model with fear effect. It is shown that, for the linear predator functional response case,the positive constant steady state is globally asymptotically stable if it ex-ists. On the other hand, for the Holling type II predator functional response case, it is proved that there exist no nonconstant positive steady states for large conversion rate. Our results limit the parameters range where complex spatiotemporal pattern formation can occur.     

Dynamical Property Analysis of a Delayed Diffusive Predator-prey Model with Fear Effect

In this paper, we study a delayed diffusive predator-prey model with fear effect and Holling II functional response. The stability of the positive equilibrium is investigated. We find that time delay can destabilize the stable equilibrium and induce Hopf bifurcation. Diffusion may lead to Turing instability and inhomogeneous periodic solutions. Through the theory of center manifold and normal form, some detailed formulas for determining the of Hopf bifurcation are presented. Some numerical simulations are also provided.     

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.     

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.     

Stability Analysis of a Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we are concerned with the dynamics of a diffusive predator-prey model that incorporates the functional response concerning hunting cooperation. First, we investigate the stability of the semi-trivial steady state. Then, we investigate the influence of the diffusive rates on the stability of the positive constant steady state. It is shown that there exists diffusion-driven Turing instability when the diffusive rate of the predator is smaller than the critical value, which is dependent on the diffusive rate of the prey, and the semi-trivial steady state and the positive constant steady state are both locally asymptotically stable when the diffusive rate of the predator is larger than the critical value. Finally, the nonexistence of nonconstant steady states is discussed.     

Second order asymptotics for the propagation speed of interfaces in the Allen-Cahn phase field model for elastic solids

For a phase field model, which consists of the elasticity equations coupled to the Allen-Cahn equation, we state an asymptotic expansion for the propagation speed of the diffusive interface. The error of the expansion is of order η², where η is the width of the interface. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of a protection zone in the diffusive Leslie predator-prey model

In this paper, we consider the diffusive Leslie predator-prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone Ω₀ for the prey is introduced. As in earlier work [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91; Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Equations 244 (2008) 61-86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over Ω₀ and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when Ω₀ is above the critical patch size. However, our research here reveals significant difference of the model's behavior from the predator-prey model studied in [Y. Du, J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations 229 (2006) 63-91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations.     

On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type

In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.     

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

Global existence of classical solutions for a class of diffusive ecological models with two free boundaries and cross-diffusion

In this paper we consider a class of diffusive ecological models with two free boundaries and with cross-diffusion and self-diffusion in one space dimension. The systems under consideration are strongly coupled, and the position of each free boundary is determined by the Stefan condition. We first show local existence of the solutions for the ecological models under some assumptions, and then prove the global existence of the solutions under extra assumptions. Our approach to the problem is by suitable changes, fixed point theorems and various estimates. Applications of these results are given to a two-species diffusive predator–prey model and a two-species diffusive competition model.     

Patterned solutions of a homogenous diffusive predator-prey system of Holling type-III

Acta Mathematicae Applicatae Sinica, English Series - In this paper, a diffusive predator-prey system of Holling type functional III is considered. For one hand, we considered the possibility of...     

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.     

Traveling wavefronts in temporally discrete reaction–diffusion equations with delay

This paper is concerned with the existence of traveling wavefronts of a temporally discrete reaction–diffusion equation with delay. By using monotone iteration and upper–lower solution technique, the existence of traveling wavefronts for the temporally discrete reaction–diffusion equation with delay is established. As an application, we consider an abstract diffusive equation, which includes a single species diffusive model as a particular case. Our result implies the temporally discrete model is a good approximation of corresponding continuous time model in sense of propagation.     

Simulation of Electrolytes in the Modeling of Batteries

Calculating the cell voltage of a lead-acid battery requires the solution of a complex multiphysical problem including electrochemical reactions on the electrode surfaces and diffusive transport in the enclosed electrolyte. Recently, the authors proposed a new electrolyte model with a thermodynamically correct coupling of mechanics to the diffusive transport. In the analysis of that model, formal asymptotic expansions were applied to handle sharp interface layers. Here, we use this analysis to formulate a simplified model for a lead-acid battery such that it is not necessary to resolve the layers numerically, but instead there are jump conditions at double layers imposed. The cell voltage is then derived using Butler-Volmer Formula. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Large Time Behavior and Selection Principle for a Diffusive Carr–Penrose Model

This paper is concerned with the study of a diffusive perturbation of the linear LSW model introduced by Carr and Penrose. A main subject of interest is to understand how the presence of diffusion acts as a selection principle, which singles out a particular self-similar solution of the linear LSW model as determining the large time behavior of the diffusive model. A selection principle is rigorously proven for a model which is a semiclassical approximation to the diffusive model. Upper bounds on the rate of coarsening are also obtained for the full diffusive model.     

Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary, II

We study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For simplicity, we assume that the environment and the solution are radially symmetric. In the special case of one space dimension and homogeneous environment, this free boundary problem was investigated in Du and Lin (2010) [10]. We prove that the spreading-vanishing dichotomy established in Du and Lin (2010) [10] still holds in the more general and ecologically realistic setting considered here. Moreover, when spreading occurs, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. When the environment is asymptotically homogeneous at infinity, these two bounds coincide. Our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension.     

Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect

In this paper, we consider travelling wave solutions for the diffusive Nicholson’s blowflies equation incorporating time delay and diffusion. Special attention is paid to the modelling of the time delay to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For the strong generic delay kernel, we show that travelling wave solutions exist provided that the delay is sufficiently small, using the geometric singular perturbation theory.     

The problem of the nonlinear diffusive predator–prey model with the same biotic resource

In recent years, the research on the diffusive predator–prey model has attracted much attention. In these models, the carrying capacity is considered as a constant. In 2013, H. M. Safuan investigated the system of a predator and prey that shares the same biotic resource, where the carrying capacity is a function of the time. The spatial component of ecological interactions has been recognized as an important factor. So, we will discuss the problem of the nonlinear diffusive predator–prey model with the same biotic resource. This model is the system of the nonlinear partial differential equations with zero-flux boundary condition. The main objective of the present paper is to investigate the existence and uniqueness of the solution of this model. In this paper, we also obtain that there is a unique solution of the nonlinear partial differential equations with Dirichlet boundary condition.     

Stability and Bifurcation Analysis in a Nonlocal Diffusive Predator-prey Model with Hunting Cooperation

In this paper, we propose a diffusive predator-prey model with hunting cooperation and nonlocal competition. Under a rather general selection of the kernel function, we first study the stability of the positive equilibrium of the model. Then, we obtain the conditions which Hopf bifurcation and Turing bifurcation occur. Our results show that nonlocal competition plays an important role in determining the dynamics of the model.