Global stability of an age-structured population model on several temporally variable patches

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number $$R_0$$ R 0 for time-independent and periodical models and establish the permanence dichotomy: if $$R_0\le 1$$ R 0 ≤ 1 , extinction on all patches is imminent, and if $$R_0>1$$ R 0 > 1 , permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.

Global Dynamics for an Age-Structured Cholera Infection Model with General Infection Rates

This paper studies the global dynamics of a cholera model incorporating age structures and general infection rates. First, we explore the existence and point dissipativeness of the orbit and analyze the asymptotical smoothness. Then, we perform rigorous mathematical analysis on the existence and local stability of equilibria. Based on the uniform persistence, we further investigate the global behavior of the cholera infection model. The results of theoretical analysis are well confirmed by numerical simulations. This research generalizes some known results and provides deeper insights into the dynamics of cholera propagation.

Bifurcation Analysis of a Diffusive SIR Model with Saturated Treatment in a Heterogeneous Environment

In this paper, we propose a diffusive SIR model with general incidence rate, saturated treatment rate and spatially heterogeneous diffusion coefficients. We first prove the global existence of bounded solutions for the model and compute the basic reproduction number. We study the local and global stabilities of the disease-free equilibrium and the uniform persistence. In the case when the diffusion rate of infected individuals is constant, we carry out a bifurcation analysis of equilibria by considering the maximal treatment rate as the bifurcation parameter. Finally, we perform some numerical simulations, which show that the solutions to our model present periodic oscillations for certain values of the parameters.

Dynamical of Ratio-Dependent Eco-epidemical Model with Prey Refuge

This paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. The prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. Here, we get the boundedness and three equilibrium points where are existence under certain conditions. In the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. We find that the system undergoes Hopf bifurcation around the interior equilibrium point by choosing as a bifurcation parameter. We also find a condition for uniform persistence. Finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge.

Effect of fear on two predator-one prey model in deterministic and fluctuating environment

Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model. Numerical examples are provided to support our obtained results.

Impact of predator fear on two competing prey species

Predator-prey interaction is a fundamental feature in the ecological system. The majority of studies have addressed how competition and predation affect species coexistence. Recent field studies on vertebrate has shown that fear of predators can influence the behavioural pattern of prey populations and reduce their reproduction. A natural question arises whether species coexistence is still possible or not when predator induce fear on competing species. Based on the above observation, we propose a mathematical model of two competing prey-one predator system with the cost of fear that affect not only the reproduction rate of both the prey population but also the predation rate of predator. To make the model more realistic, we incorporate intraspecific competition within the predator population. Biological justification of the model is shown through positivity and boundedness of solutions. Existence andstability of different boundary equilibria are discussed. Condition for the existence of coexistence equilibrium point is derived from showing uniform persistence. Local as well as a global stability criterion is developed. Bifurcation analysis is performed by choosing the fear effect as the bifurcation parameter of the model system. The nature of the limit cycle emerging through a Hopf bifurcation is indicated. Numerical experiments are carried out to test the theoretical results obtained from this model.

Uniform Persistence and Periodic Solutions of Generalized Predator–Prey Type Eco-Epidemiological Systems

In this paper, we analyze a class of three-dimensional eco-epidemiological models where prey is subject to Allee effects and infection. We first establish the existence, uniqueness, positivity and uniform ultimate boundedness of the solutions for the proposed system in the positive octant. For three subsystems, we investigate the existence of their respective trivial and positive equilibria and determine the conditions for some bifurcations (Hopf bifurcation, Bogdanov–Takens bifurcation of codimension-2 and saddle-node bifurcation) to occur. We find that the Allee effect, nonmonotonic functional response and intra-class competition in susceptible preys enable the S–I and S–P subsystems to have richer dynamics. For example, the S–I subsystem can have up to three positive equilibria, the S–P subsystem with nonmonotonic functional response can have two positive equilibria while it is impossible in monotonic situation, and high intra-class competition in susceptible preys may lead to the extinction of the predator population, etc. We show that the strong Allee effect can create a separatrix curve (or surface), leading to multistability. Then, we study the uniform persistence of the full system and identify an interior periodic orbit by applying Poincaré map and bifurcation theory. Our analysis reveals that the introduction of the infection or predation may act as a biological control to save the population from extinction and the interaction between these two factors yields a diverse array of biologically relevant behaviors. Finally, some numerical simulations are performed to support and supplement our analytical findings.

Mathematical analysis of an HTLV-I infection model with the mitosis of CD4+ T cells and delayed CTL immune response

In this paper, we consider an improved Human T-lymphotropic virus type I (HTLV-I) infection model with the mitosis of CD4+ T cells and delayed cytotoxic T-lymphocyte (CTL) immune response by analyzing the distributions of roots of the corresponding characteristic equations, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium when the CTL immune delay is zero is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. We define the immune-inactivated reproduction ratio R0 and the immune-activated reproduction ratio R1. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that if R0 < 1, the infection-free equilibrium is globally asymptotically stable; if R1 < 1 < R0, the immunity-inactivated equilibrium is globally asymptotically stable; if R1 > 1, the immunity-activated equilibrium is globally asymptotically stable when the CTL immune delay is zero. Besides, uniform persistence is obtained when R1 > 1. Numerical simulations are carried out to illustrate the theoretical results.

Global dynamics analysis of a Zika transmission model with environment transmission route and spatial heterogeneity

<abstract><p>Zika virus, a recurring mosquito-borne flavivirus, became a global public health agency in 2016. It is mainly transmitted through mosquito bites. Recently, experimental result demonstrated that $ Aedes $ mosquitoes can acquire and transmit Zika virus by breeding in contaminated aquatic environments. The environmental transmission route is unprecedented discovery for the Zika virus. Therefore, it is necessary to introduce environment transmission route into Zika model. Furthermore, we consider diffusive terms in order to capture the movement of humans and mosquitoes. In this paper, we propose a novel reaction-diffusion Zika model with environment transmission route in a spatial heterogeneous environment, which is different from all Zika models mentioned earlier. We introduce the basic offspring number $ R_{0}^{m} $ and basic reproduction number $ R_{0} $ for this spatial model. By using comparison arguments and the theory of uniform persistence, we prove that disease free equilibrium with the absence of mosquitoes is globally attractive when $ R_{0}^{m} &lt; 1 $, disease free equilibrium with the presence of mosquitoes is globally attractive when $ R_{0}^{m} &gt; 1 $ and $ R_{0} &lt; 1 $, the model is uniformly persistent when $ R_{0}^{m} &gt; 1 $ and $ R_{0} &gt; 1 $. Finally, numerical simulations conform these analytical results.</p></abstract>