Jian Zu

The impact of Allee effect on a predator–prey system with Holling type II functional response

Abstract In this paper, the Allee effect is incorporated into a predator–prey model with Holling type II functional response. Compared with the predator–prey model without Allee effect, we find that the Allee effect of prey species increases the extinction risk of both predators and prey. When the handling time of predators is relatively short and the Allee effect of prey species becomes strong, both predators and prey may become extinct. Moreover, it is shown that the model with Allee effect undergoes the Hopf bifurcation and heteroclinic bifurcation. The Allee effect of prey species can lead to unstable periodical oscillation. It is also found that the positive equilibrium of the model could change from stable to unstable, and then to stable when the strength of Allee effect or the handling time of predators increases continuously from zero, that is, the model admits stability switches as a parameter changes. When the Allee effect of prey species becomes strong, longer handling time of predators may stabilize the coexistent steady state.

GLOBAL DYNAMICS OF A MULTI-GROUP EPIDEMIC MODEL WITH GENERAL RELAPSE DISTRIBUTION AND NONLINEAR INCIDENCE RATE

In this paper, we investigate a class of multi-group epidemic models allowing heterogeneity of the host population and that has taken into consideration with general relapse distribution and nonlinear incidence rate. We establish that the global dynamics are completely determined by the basic reproduction number R0. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals and a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach. Biologically, the disease (with any initial inoculation) will persist in all groups of the population and will eventually settle at a constant level in each group. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behaviors of the basic SIR model. On the other hand, the global dynamics exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.

The evolution of phenotypic traits in a predator–prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.

The independent impact of newborn hepatitis B vaccination on reducing HBV prevalence in China, 1992–2006: A mathematical model analysis

OBJECTIVE To evaluate the independent impact of newborn hepatitis B vaccination on reducing HBV prevalence in China, from its introduction in 1992 to 2006. METHODS An age- and time-dependent discrete dynamic model was developed to simulate HBV transmission in China under the assumptions of no any change in interventions and only with newborn vaccination introduction, respectively. The initial conditions of the model were determined according to the national serosurvey in 1992. The simulated results were compared with the observed results of the national serosurvey in 2006, and the contribution rate of newborn vaccination on reducing HBV prevalence was calculated overall and by birth cohort. RESULTS The total HBV prevalence would remain stable through the 14-year period if no any change in interventions, but decrease year by year if only with newborn vaccination introduction. Newborn vaccination could account for more than 50% of the reduction of the total HBV prevalence, although the full 3-dose and timely birth dose vaccination coverage rates were low in the early years. The results by birth cohort showed that the higher the two coverage rates, the higher contribution rate on reducing HBV prevalence. For the 2005 birth cohort which had high levels in the two coverage rates, the contribution rate could reach more than 95%. CONCLUSION Newborn hepatitis B vaccination from 1992 to 2006 in China had played the most important role in reducing HBV prevalence. Newborn vaccination with high full 3-dose and timely birth dose coverage rates is the decisive factor in controlling hepatitis B in China.

On evolution under symmetric and asymmetric competitions

This paper considers the coevolution of phenotypic traits in a community comprising two competitive species subject to strong Allee effects. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy under symmetric competition. Secondly, we find that evolutionary suicide is impossible when the two species undergo symmetric competition, however, evolutionary suicide can occur in an asymmetric competition model with strong Allee effects. Thirdly, it is found that evolutionary bistability is a likely outcome of the process under both symmetric and asymmetric competitions, which depends on the properties of symmetric and asymmetric competitions. Fourthly, under asymmetric competition, we find that evolutionary cycle is a likely outcome of the process, which depends on the properties of both intraspecific and interspecific competition. When interspecific and intraspecific asymmetries vary continuously, we also find that the evolutionary dynamics may admit a stable equilibrium and two limit cycles or two stable equilibria separated by an unstable limit cycle or a stable equilibrium and a stable limit cycle.

Adaptive evolution of foraging-related traits in a predator–prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.

Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis

On the ecological timescale, two predator species with linear functional responses can stably coexist on two competing prey species. In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate under what conditions such a coexistence is also evolutionarily stable, and whether the two predator species may evolve from a single ancestor via evolutionary branching. We assume that predator strategies differ in capture rates and a predator with a high capture rate for one prey has a low capture rate for the other and vice versa. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly convex in the vicinity of the singular strategy and the interspecific prey competition is not strong, then this singular strategy is an evolutionary branching point, near which the resident and mutant predator populations can coexist and diverge in their strategies. Second, we find that after branching has occurred in the predator phenotype, if the trade-off curve is globally convex, the predator population will eventually branch into two extreme specialists, each completely specializing on a particular prey species. However, in the case of smoothed step function-like trade-off, an interior dimorphic singular coalition becomes possible, the predator population will eventually evolve into two generalist species, each feeding on both of the two prey species. The algebraical analysis reveals that an evolutionarily stable dimorphism will always be attractive and that no further branching is possible under this model.

Adaptive evolution of attack ability promotes the evolutionary branching of predator species.

In this paper, with the methods of adaptive dynamics and critical function analysis, we investigate the evolutionary branching phenomenon of predator species. We assume that both the prey and predators are density-dependent and the predators attack ability can adaptively evolve, but this has a cost in terms of its death rate. First, we identify the general properties of trade-off relationships that allow for a continuously stable strategy and evolutionary branching in the predator strategy. It is found that if the trade-off curve is weakly concave near the singular strategy, then the singular strategy may be an evolutionary branching point. Second, we find that after the branching has occurred in the predator strategy, if the trade-off curve is convex-concave-convex, the predator species will eventually evolve into two different types, which can stably coexist on the much longer evolutionary timescale and no further branching is possible.

Adaptive evolution of anti-predator ability promotes the diversity of prey species: Critical function analysis

In this paper, with the method of adaptive dynamics and critical function analysis, we investigate the evolutionary diversification of prey species. We assume that prey species can evolve safer strategies such that it can reduce the predation risk, but this has a cost in terms of its reproduction. First, by using the method of critical function analysis, we identify the general properties of trade-off functions that allow for continuously stable strategy and evolutionary branching in the prey strategy. It is found that if the trade-off curve is globally concave, then the evolutionarily singular strategy is continuously stable. However, if the trade-off curve is concave-convex-concave and the preys sensitivity to crowding is not strong, then the evolutionarily singular strategy may be an evolutionary branching point, near which the resident and mutant prey can coexist and diverge in their strategies. Second, we find that after branching has occurred in the prey strategy, if the trade-off curve is concave-convex-concave, the prey population will eventually evolve into two different types, which can coexist on the long-term evolutionary timescale. The algebraical analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible for the concave-convex-concave trade-off relationship.

Top predators induce the evolutionary diversification of intermediate predator species.

We analyze the evolutionary branching phenomenon of intermediate predator species in a tritrophic food chain model by using adaptive dynamics theory. Specifically, we consider the adaptive diversification of an intermediate predator species that feeds on a prey species and is fed upon by a top predator species. We assume that the intermediate predator׳s ability to forage on the prey can adaptively improve, but this comes at the cost of decreased defense ability against the top predator. First, we identify the general properties of trade-off relationships that lead to a continuously stable strategy or to evolutionary branching in the intermediate predator species. We find that if there is an accelerating cost near the singular strategy, then that strategy is continuously stable. In contrast, if there is a mildly decelerating cost near the singular strategy, then that strategy may be an evolutionary branching point. Second, we find that after branching has occurred, depending on the specific shape and strength of the trade-off relationship, the intermediate predator species may reach an evolutionarily stable dimorphism or one of the two resultant predator lineages goes extinct.