Narrow Results

In this work, we consider a stage-structured cannibalism model with two delays. One delay characterizes the lag effect of negative feedback of the prey species, the other has the effect of gestation of the adult predator population. Firstly, criteria for the local stability of feasible equilibria are established. Meanwhile, by choosing delay as a bifurcation parameter, the criteria on the existence of Hopf bifurcation are established. Furthermore, by the normal form theory and center manifold theorem, we derive the explicit formulas determining the properties of periodic solutions. Finally, the theoretical results are illustrated by numerical simulations, from which we can see that the predator’s gestation time delay can make the chaotic phenomenon disappear and maintain periodic oscillation, and that a large feedback time delay of prey can make predators extinct and prey form a periodic solution.

A stage-structured predator-prey model concerning impulsive control strategy is proposed and investigated. The global attractiveness of the pest-eradication periodic solution is discussed, and sufficient conditions are obtained for the permanence of the system. Further, numerical simulations show that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system with constant periodic impulsive perturbations admits rich and complex dynamics.

Age structured models are important for the dynamics and evolution of many insect populations. For such models the rates of survival, growth and reproduction depend on the age or developmental stage of individuals in the population. For example, the life cycles of many species are composed of at least two stages, immature individuals (immatures) and mature individuals (matures) which may exhibit fundamentally different developmental, morphological and behavioral characteristics. In this paper, we incorporate these biological properties into a predator–prey population model. Our model is stage-structured for the prey and contains maturation and gestation delays. Using comparison theorems, we derive sufficient conditions for dynamical system permanence. We found equilibrium points exhibit switching with boundary equilibrium points through a mechanism of decreasing maturation of the prey population. Furthermore, an analysis of the corresponding characteristic equations was performed to determine local stability of equilibria. Choosing the gestation delay as a bifurcation parameter, we were able to compute a critical value for the existence of small amplitude oscillation of population densities (i.e., a Hopf bifurcation). Sufficient conditions for the global stability of all non-negative equilibria were obtained using Kamke comparison theorems and a newly developed iteration technique. Numerical simulations were performed to illustrate the theoretical results.

In this paper, a harvesting fishery model in a two-patch environment: one free-fishing zone and the other one reserved zone where fishing is strictly prohibited, is proposed and analyzed. The existence of possible biological steady states, along with their local stability, instability and global stability is discussed. The existence of bioeconomic equilibrium is derived. An optimal harvesting policy is also given by applying pontryagin's maximum principle.

We investigate the global behavior for a population whose members are divided into two classes: juveniles and adults, with mortalities age- and density-dependent. Laplace transforms are used to examine the asymptotic dynamics of age profiles, which describes completely the evolution process of the nonlinear population system. Our result can be viewed as a generalization of Sharp–Lotka theorem in some nonlinear situations.

In this paper, we considered a time-delay predator–prey system, in which the prey has two life stages, juvenile and mature. Delay was regarded as the bifurcation parameter, we analyzed the characteristic equation of the system at the positive equilibrium, stability of the positive equilibrium and existence of Hopf bifurcation with delay τ in the term of degree are investigated. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. To verify our theoretical results, a numerical example is also included.

In this paper, a stage-structured predator–prey model is proposed and analyzed with density-dependent maturation delay. We studied the dynamics of our model analytically and obtained conditions which influence the positivity and boundedness of all populations. Criteria for the existence of a non-trivial equilibrium and conditions for the uniqueness of this equilibrium are given. A linearized analysis on the equilibria, which is algebraically very complicated in the case of non-trivial equilibrium, is carried out. We proved that the system is globally asymptotically stable in the situation when non-trivial equilibrium does not exist. To accomplish our all analytical findings and to investigate the effect of density-dependent maturation delay on the system behavior, we presented a numerical simulation. It is concluded that variations in parameter, which we introduce in the system to observe the effect of density-dependent maturation delay, produces significant quantitative changes in system behavior and also qualitative changes in the behavior of immature predator population.

In this paper, a stage-structured predator–prey system with birth pulse and disturbed time delay is investigated. The conditions of the prey-extinction periodic solution of the system which are globally attractive have been obtained. Furthermore, the sufficient conditions for the permanence of the system are established. Finally, numerical analysis is given to confirm the theoretical results.

A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle's invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.

In this paper, a class of brucellosis transmission model with seasonal alternation, density-dependent growth, stage-structure, maturation delay, time-varying incubation is established. The basic reproduction number ${\mathcal{ℛ}}_0$ is derived, by which we find that the brucellosis is uniformly persistent if ${\mathcal{ℛ}}_0>1$, while the disease-free periodic solution is globally attractive if ${\mathcal{ℛ}}_0<1$. The theoretical results are illustrated by numerical simulation, from which we find that the brucellosis transmission would be overestimated (or underestimated) if we ignore the influence of time-varying incubation or maturation delay. If density-dependent growth of animals is ignored, the risk of brucellosis may be far underestimated, the extinction of brucellosis can be obtained by numerical simulation under the same conditions. Seasonality significantly affects the long-term dynamic behavior of brucellosis, and the inconsistency of parameter periods results in complex dynamic behavior.

In biological pest control systems, several pests (including insects, mites, weeds, etc.) are controlled by biocontrol agents that rely primarily on predation. Following this biocontrol management ecology, we have created a three-tier prey–predator model with prey phase structure and predator gestation delay. Several studies have demonstrated that predators with Holling type-II functional responses sometimes consume immature prey. A study of the well-posedness and local bifurcation (such as saddle-node and transcritical) near the trivial and planer equilibrium points is carried out. Without any time lag, the prey development coefficient has a stabilizing impact, while increasing attack rate accelerates instability. Energy transformation rate and handling time are shown to cause multiple stability switches in the system. Numerical results demonstrate time delay is the key destabilizer that destroys stability. Our model can replicate more realistic events by including time-dependent factors and exploring the dynamic behavior of nonautonomous systems. In the presence of time delay, sufficient conditions of permanence and global attractivity of the nonautonomous system are derived. Finally, MATLAB simulations are performed to validate the analytical findings.

In this paper, by using the continuation theorem of coincidence degree theory, a sufficient condition of existence of positive periodic solutions is obtained for an stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response. By constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Our result is a good complement to the earlier publications.