Existence and asymptotic behavior of solutions for a mathematical ecology model with herd behavior

In this work, we consider a prey-predator model with herd behavior under Neumann boundary conditions. For the system without diffusion, we establish a sufficient condition to guarantee the local asymptotic stability of all nontrivial equilibria and prove the existence of limit cycle of our proposed model. For the system with diffusion, we consider the long time behavior of the model including global attractor and local stability, and the Hopf and steady-state bifurcation analysis from the unique homogeneous positive steady state are carried out in detail. Furthermore, some numerical simulations to illustrate the theoretical analysis are performed to expand our theoretical results.     

The stability of a diffusion model of plankton allelopathy with spatio–temporal delays

A reaction–diffusion system with non-local delay is proposed to describe two competitive planktonic growths in aquatic ecology. The local and global stability of the axial equilibria as well as the positive equilibrium are discussed. Our results show that the delay has no effect on the stability of the axial equilibria; on the other hand, the positive equilibrium can be induced to be locally unstable by the delay. Finally, the corresponding numerical simulations are also demonstrated.     

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.     

Traveling wavefronts of a prey-predator diffusion system with stage-structure and harvesting

From a biological point of view, we consider a prey-predator-type free diffusion fishery model with stage-structure and harvesting. First, we study the stability of the nonnegative constant equilibria. In particular, the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation. Then, employing the upper and lower solution method, we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system. Numerical simulation is also carried out to illustrate the main result.     

THE EFFECT OF A PREDATOR RESOURCE ON A DIFFUSIVE PREDATOR PREY SYSTEM

A model of a predator-prey system with diffusion and predator resource is studied. Both constant and variable resources are considered. In the absence of diffusion, criteria for local stability, instability, and global stability of equilibria, as well as persistence and extinction, are obtained. It is shown that an otherwise unstable uniform equilibrium state may be stabilized by diffusion.     

Delay induced subcritical Hopf bifurcation in a diffusive predator-prey model with herd behavior and hyperbolic mortality

In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Finally, some numerical simulations are also carried out and we obtain the unstable spatial periodic solutions, which are induced by the subcritical Hopf bifurcation.     

Hopf Bifurcations in a Predator-Prey Diffusion System with Beddington-DeAngelis Response

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

Stability of a three-dimensional diffusive Lotka–Volterra system of type-K with delays

A three-dimensional diffusive Lotka–Volterra system of type-K with delays is investigated. We give a stability analysis in detail for all equilibria of the system and obtain some threshold conditions for linear instability and linear asymptotic stability of each equilibrium. We develop the analytical method for stability analysis of reaction–diffusion equations with multi-delays.     

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations.     

Dynamics of an epidemic model with relapse and delay

In this paper, we consider a new epidemiological model with delay and relapse phenomena. Firstly, a basic reproduction number $R_0$ is identified, which serves as a threshold parameter for the stability of the equilibria of the model. Then, beginning with the delay-free model, the global asymptotic stability of the equilibria is obtained through the construction of suitable Lyapunov functions. For the delay model, the stability of the positive equilibrium and the existence of the local Hopf bifurcation are discussed. Furthermore, the application of the normal form theory and center manifold theorem is used to determine the direction and stability of these Hopf bifurcations. Finally, we shed light on corresponding biological implications from a numerical perspective. It turns out that time delay affects the stability of the positive equilibrium, leading to the occurrence of periodic oscillations and disease recurrence.     

Global behavior of solutions in a Lotka–Volterra predator–prey model with prey-stage structure

In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large.     

Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment

In this paper, the dynamics behavior of a delayed viral infection model with logistic growth and immune impairment is studied. It is shown that there exist three equilibria. By analyzing the characteristic equations, the local stability of the infection-free equilibrium and the immune-exhausted equilibrium of the model are established. By using suitable Lyapunov functional and LaSalle invariant principle, it is proved that the two equilibria are globally asymptotically stable. In the following, the stability of the positive equilibrium is investigated. Furthermore, we investigate the existence of Hopf bifurcation by using a delay as a bifurcation parameter. Finally, numerical simulations are carried out to explain the mathematical conclusions.     

Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.     

Turing patterns and spatiotemporal patterns in a tritrophic food chain model with diffusion

In this paper, the temporal, spatial, and spatiotemporal patterns of a tritrophic food chain reaction–diffusion model with Holling type II functional response are studied. Firstly, for the model with or without diffusion, we perform a detailed stability and Hopf bifurcation analysis and derive criteria for determining the direction and stability of the bifurcation by the center manifold and normal form theory. Moreover, diffusion-driven Turing instability occurs, which induces spatial inhomogeneous patterns for the reaction–diffusion model. Then, the existence of positive non-constant steady-states of the reaction–diffusion model is established by the Leray–Schauder degree theory and some a priori estimates. Finally, numerical simulations are presented to visualize the complex dynamic behavior.     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.     

Dynamics of product inhibition on lactic acid fermentation

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincare-Bendixson theorem, we obtain the condition for the globally asymptotical stability of the equilibria. Secondly, using the Floquet’s theorem and small-amplitude perturbation method, we obtain the biomass-free periodic solution is locally stable if R₂ < 1. The permanent conditions of the system are also given. Finally, our findings are confirmed by means of numerical simulations.     

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

In this paper, a diffusive predator–prey system with Holling III functional response and nonconstant death rate subject to Neumann boundary condition is considered. We study the stability of equilibria, and Turing instability of the positive equilibrium. We also perform a detailed Hopf bifurcation analysis to PDE system, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. In addition, some numerical simulations are carried out.     

Numerical detection of instability regions for delay models with delay-dependent parameters

In this paper we are interested in gaining local stability insights about the interior equilibria of delay models arising in biomathematics. The models share the property that the corresponding characteristic equations involve delay-dependent coefficients. The presence of such dependence requires the use of suitable criteria which usually makes the analytical work harder so that numerical techniques must be used. Most existing methods for studying stability switching of equilibria fail when applied to such a class of delay models. To this aim, an efficient criterion for stability switches was recently introduced in [E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002) 1144–1165] and extended [E. Beretta, Y. Tang, Extension of a geometric stability switch criterion, Funkcial Ekvac 46(3) (2003) 337–361]. We describe how to numerically detect the instability regions of positive equilibria by using such a criterion, considering both discrete and distributed delay models.