Coexistence periodic solutions of a doubly nonlinear parabolic system with Neumann boundary conditions

This paper is concerned with a competitive and cooperative mathematical model for two biological populations which dislike crowding, diffuse slowly and live in a common territory under different kind of intra- and inter-specific interferences. The model consists of a system of two doubly nonlinear parabolic equations with nonlocal terms and Neumann boundary conditions. Based on the theory of the Leray–Schauder degree, we obtain the coexistence periodic solutions, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two interacting populations, under different intra- and inter-specific interferences on their natural growth rates.     

Permanence,extinction and periodic solutions in a mathematical model of cell populations affected by periodic radiation

A periodic mathematical model of cell populations affected by periodic radiation is presented and studied in this paper. We obtain some sufficient conditions on the permanence and extinction of the system. Furthermore, criteria on the existence and global asymptotic stability of unique positive periodic solutions are established. Some numerical examples are shown to verify our results. A discussion is presented for further study.     

Impulsive harvesting and by-catch mortality for the theta logistic model

In this paper, we investigate the population dynamics described by the theta logistic model with periodic impulsive harvesting and by-catch mortality. We examine the existence and stability of two positive periodic solutions by using qualitative methods and cobwebs. Then the sufficient conditions under which the unique positive periodic solution exists and is semi-stable are established, and qualifications for the solutions approach zero are also obtained. Further, choosing the maximum sustainable yield as the management objective, we investigate the optimal harvesting policy for the theta logistic model with periodic impulsive harvesting. Moreover the corresponding theta logistic difference equation is considered subject to the impulsive perturbation, and the dynamics which is parallel to that for the differential equation is examined. The main results extend and generalize the classical results for populations described by the autonomous logistic equation in renewable resources management.     

Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

SUSTAINABLE FOREST MANAGEMENT FOR OPTIMIZING MULTISPECIES WILDLIFE HABITAT: A COASTAL DOUGLAS-FIR EXAMPLE

Wildlife species viability optimization models are developed to convert a given set of initial forest conditions, through a combination of natural growth and management treatments, to a forest system which addresses the joint habitat needs of multispecies populations over time. A linear model of forest cover and wildlife populations is used to form a system of forest management control variables for wildlife habitat modification. The paper examines two objective functions coupled to this system for optimizing sustainable joint species viability. The first maximizes the product of periodic joint viabilities over all time periods, focusing management resources on long-term equilibria, with less emphasis on conversion strategy. The second iteratively maximizes the minimum periodic joint viability over all time periods. This focuses management resources on the most limiting time periods, typically the conversion phase periods. Both objective functions resulted in either point or cyclic equilibria, with cycle lengths equal to minimum forest treatment ages. A third objective, based on maximizing the minimum individual species periodic viability is used to examine single species emphasis. Examples are developed through a case study of 92 vertebrate species found in coastal Douglas-fir stands of northwestern California.     

Hawk-dove game and competition dynamics

In this article, we consider two populations subdivided into two categories of individuals (hawks and doves). Individuals fight to have access to a resource necessary for their growth. Conflicts occur between hawks of the same population and hawks of different populations. The aim of this work is to investigate the long term effects of these conflicts on coexistence and stability of the community of the two populations. This model involves four variables corresponding to the two tactics of individuals of the two populations. The model is composed of two parts, a fast part describing the encounters and fights, and the slow part describing the long term effects of encounters on the growth of the populations. We use aggregation methods allowing us to reduce this model into a system of two ODEs for the total densities of the two populations. This is found to be a classical Lotka-Volterra competition model. We study the effects of the different fast equilibrium proportions of hawks and doves in both populations on the global coexistence and the mutual exclusion of the two populations. We show that in some cases, mixed hawk and dove populations coexist. Aggressive populations of hawks exclude doves except in the case of interpopulation costs being smaller than intrapopulation ones.     

Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely without human intervention and derive conditions for this to occur. In this case, the system of impulsive differential equations has a periodic solution. We show that success is equivalent to the convergence of solutions to this periodic solution. We provide conditions that ensure that a periodic solution exists. When it exists, it is unique and attracting. However, we also show that whether a solution converges to this periodic solution, and hence whether the model predicts that the reactor operates successfully, is initial condition dependent. The analysis is illustrated with numerical examples.     

Periodic and Almost Periodic Solutions for a Coupled System of Two Korteweg-de Vries Equations with Boundary Forces

In this paper, we investigate a coupled system of two Korteweg-de Vries equations on a bounded domain. We discuss the long-time behavior of this system with forces on the left Dirichlet boundary conditions. We obtain that if the forces are periodic (almost periodic) with small amplitude, then the solution of the coupled system is periodic (almost periodic).     

比率型-捕食者-两竞争食饵模型的动力学行为

本文研究比率型非自治的捕食者 -食饵模型 .该系统是两个具有竞争关系的食饵种群被一个捕食种群捕食 .我们研究其动力学行为 ,包括持久性 ,全局渐近稳定性 ,周期解 ,概周期解的存在唯一性     

Chaos in periodically forced discrete-time ecosystem models

Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran–Ricker, and Hassell, and also a Maynard Smith predator–prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.     

Using a signature function to determine resonant and attenuant 2-cycles in the Smith–Slatkin population model

We study the responses of discretely reproducing populations to periodic fluctuations in three parameters: the carrying capacity and two demographic characteristics of the species. We prove that small 2-periodic fluctuations of the three parameters generate 2-cyclic oscillations of the population. We develop a signature function for predicting the responses of populations to 2-periodic fluctuations. Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the three parameters. Periodic environments are deleterious for populations when the signature function is negative, while positive signature functions signal favorable environments. We compute the signature function for the Smith–Slatkin model, and use it to determine regions in parameter space that are either favorable or detrimental to the species.     

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Study on autonomous and nonautonomous version of a food chain model with intraspecific competition in top predator

In this article, a three-species food chain model with intraspecific competition in top predators has been considered. Ecological and mathematical well posedness of the model system has been established by showing that all the solutions of the model are positive and bounded. The extinction scenarios of intermediate and top predator species along with the existence and stability of all equilibrium points have been discussed. The effects of competition and conversion efficiency of top predators in the dynamics of the system have been discussed with great thrust, and it is observed that the conversion efficiency of top predators deteriorates the stability of the system, whereas intraspecific competition in top predators enhances the stable coexistence of all the populations of the system. Further, nonautonomous version of the model system has been taken into consideration to study the impact of seasonal variation in the dynamics of the model system. Sufficient conditions for the existence of a globally attractive positive periodic have been established in a periodic environment. Finally, numerical simulations have been carried out to validate our analytical findings.     

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.     

Hybrid approach for pest control with impulsive releasing of natural enemies and chemical pesticides: A plant–pest–natural enemy model

The agricultural pests can be controlled effectively by simultaneous use (i.e., hybrid approach) of biological and chemical control methods. Also, many insect natural enemies have two major life stages, immature and mature. According to this biological background, in this paper, we propose a three tropic level plant–pest–natural enemy food chain model with stage structure in natural enemy. Moreover, impulsive releasing of natural enemies and harvesting of pests are also considered. We obtain that the system has two types of periodic solutions: plant–pest-extinction and pest-extinction using stroboscopic maps. The local stability for both periodic solutions is studied using the Floquet theory of the impulsive equation and small amplitude perturbation techniques. The sufficient conditions for the global attractivity of a pest-extinction periodic solution are determined by the comparison technique of impulsive differential equations. We analyze that the global attractivity of a pest-extinction periodic solution and permanence of the system are evidenced by a threshold limit of an impulsive period depending on pulse releasing and harvesting amounts. Finally, numerical simulations are given in support of validation of the theoretical findings.     

Global analysis of a model of differential susceptibility induced by genetics

This paper is concerned with global analysis of an SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals. We prove that this system has no periodic solutions and use the Poincaré index theorem to determine the number of rest points and their stability properties. It has been shown that multiple equilibria (bistability) occurs for suitable values of parameters. We also give some numerical examples of all possible bifurcations of this system.     

The dynamics of an age structured predator–prey model with disturbing pulse and time delays

Many of the existing predator–prey models on stage structured populations are some ordinary differential equations (ODE) or models without a disturbing effect of human behavior. In reality, death of the juvenile during its immature stage and catching or poisoning for the prey or predator occur continuously. From this basic standpoint, we formulate a general and robust prey-dependent consumption predator–prey model with periodic harvesting (catching or poisoning) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. We show that the conditions for global attractivity of the ‘predator-extinction’ (‘predator-eradication’) periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. We also show that constant maturation time delay and impulsive catching or poisoning for the prey can bring great effects on the dynamics of system by numerical analysis. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with age structure, exhibit a new modeling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat

In this paper, we introduce and study a model of a predator–prey system with Monod type functional response under periodic pulsed chemostat conditions, which contains with predator, prey, and periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

On periodic Gibbs measures of <Emphasis Type="Italic">p</Emphasis>-adic Potts model on a Cayley tree

In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.