Zizhen Li

Effects of prey refuges on a predator-prey model with a class of functional responses : The role of refuges

In this paper, the effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using the analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We will evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field.

Global stability of an SIR epidemic model with constant infectious period

In this paper, we derive and study an SIR epidemic model with constant infectious period which is incorporated as a time delay. Both trivial and endemic equilibrium are found, and their stability is investigated. Using Lyapunov functional approach, sufficient conditions for global stability of endemic equilibrium is obtained.

Dynamical complexity and metapopulation persistence

Factors leading to chaotic dynamics in metapopulation are studied, including Allee effect, rescue effect and overcrowding effect. We use theoretical bifurcation diagram and lattice simulation to investigate the dynamics of metapopulation in time and space. The influence of habitat destruction on the complexity of dynamics, which can decrease the fraction of suitable patches and induce the change of colonization rate and extinction rate, is also discussed. The whole scene of the implicit relationship between dynamical complexity and metapopulation persistence is given, which has five primary results. First, metapopulation persistence reaches the maximum at moderate dynamical complexity, which can occur at moderate intensity of habitat destruction. Second, the intensity of habitat destruction can be approximated by the dynamical complexity because the habitat destruction regulates the dynamics. Third, Allee effects can amazingly regulate dynamics and improve the persistence slightly before extinction. Fourth, rescue effects stabilize dynamics and improve persistence. Finally, overcrowding effect is the key to incur chaos and increase the dynamical complexity, and to improve the persistence of metapopulations. Anyway, there is no uniform correlation between dynamical complexity and metapopulation persistence.

Distribution patterns of metapopulation determined by Allee effects

The Allee effect, the social dysfunction and failure to mate successfully when population density falls below a certain threshold, is one of the most important phenomena in ecology that can profoundly affect metapopulation persistence. We have developed a continuous dynamic model by pair approximation and two derived spatial lattice models to describe the influences of Allee effects on the distribution and dynamics of metapopulation. Analytical results of pair approximation show that the initial global stable equilibrium of metapopulation size turns into a local stable equilibrium with Allee effects and sensitivity to the initial situations that can incur a threshold phenomenon in dynamics. When the intensity of the Allee effect varies within a certain range, a new positive local stable equilibrium appears. This new equilibrium has the same local intensity as the initial one and a smaller metapopulation size. However, a metapopulation with a too strong Allee effect is doomed. Simulation results from the lattice models reinforce these findings and show that the new equilibrium forms a static distribution border in space. Hence, an Allee effect with moderate intensity can incur three distribution patterns that are sensitive to the initial metapopulation size and the spatial configuration of local populations: aggregation, circumscription and extinction. The pattern of circumscription may be a new explanation for the species’ current distributional range. The relationships between distribution patterns (such as random, uniform, aggregation, circumscription and extinction) and other factors (such as mean-field assumption, local interaction, demographic stochasticity and Allee effect) are also discussed.

Influence of intraspecific density dependence on a three-species food chain with and without external stochastic disturbances

Abstract Natural population dynamics are very complicated. Previous theory has shown that this phenomenon can be explained by the stochastic behavior arising from stochastic disturbances of external environment factors and the highly complex and chaotic behavior generating from nonlinearities in population itself, which is considered as a leading one. However, recently many studies theoretically show that ecological factors such as immigration, omnivory, habitat-heterogeneity may impede and control chaos. In this paper, we investigate the role of intraspecific density dependence (IDD), another important ecological factor, in the dynamics of two versions (deterministic and stochastic) of a food chain. We find that the addition of IDD to a deterministic three-species food chain model stabilizes the food chain system, leading chaotic and periodic dynamics to a steady state and especially making the very famous ‘teacup’ chaotic attractor disappear, and that the addition of it to a stochastic one at most results in a reduction in amplitude and frequency of dynamical fluctuations and can never eliminate the stochastic behavior. Our results contribute to investigations of the relative importance of the intrinsic factors and extrinsic environmental factors in determining population size fluctuations. Also, they give a better understanding of the essential difference between chaos and stochasticity.

Population dynamics and the color of environmental noise: A study on a three-species food chain system

In the present study, the effect of ‘red’, ‘white’ and ‘blue’ environmental noise on population dynamics in a simple three-species food chain system was analyzed. The ‘colored’ noise was superimposed on the three-species food chain model first put forth by Rosenzweig in different ways and the resulting power spectra were investigated. We showed that the amplitude of environmental noise, the trophic level at which a population is positioned and whether a population is directly affected by environmental noise, are all important with respect to the way in which a population responds to noise with different colors. For the deterministic case, all population dynamics are ‘red’ irrespective of the system dynamics. When all species are sensitive to environmental noise, the top predator’s dynamics always remain ‘red’ regardless of the color of the noise and its amplitude, whereas the dynamics of the intermediate species turn ‘blue’ under disturbances of any color with high amplitude, and those of the basal species may become ‘blue’ only under ‘blue’ noise. If only one species is sensitive to environmental noise, the dynamics of the insensitive species are always ‘red’, irrespective of the color of the noise and its amplitude. Unlike previous results obtained by studying single-species models, our results have almost nothing to do with deterministic system dynamics. In other words, changing the deterministic system dynamics from stable via periodic to chaotic does not qualitatively change the outcome. Our results are of importance in determining how we interpret the ubiquitous ‘red’ power spectrum of natural ecological time series.

A model of ecosystem health and its application

Abstract Aimed at the special background of arid region Shapotou in China, the paper presented a measure of ecosystem health, which was based on three criteria proposed by Costanza: vigor, organization, and resilience, then modified Costanza’s model, and calculated. The calculated results show that the mixed species plots had the highest overall health, this means that the monitoring and managing modes of mixed species plots are more suitable. The conclusion provide quantitative basis for restoration and rehabilitation of artificial vegetation in arid and semi-arid regions.

The effect of landscape heterogeneity on host-parasite dynamics

Environmental heterogeneity has been shown to have a profound effect on population dynamics and biological invasions, yet the effect of its spatial structure on the dynamics of disease invasion in a spatial host–parasite system has received little attention. Here we explore the effect of environment heterogeneity using the pair approximation and the stochastic spatially explicit simulation in which the lost patches are clustered in a fragmented landscape. The intensity of fragmentation is defined by the amount and spatial autocorrelation of the lost habitat. More fragmented landscape (high amount of habitat loss, low clustering of lost patches) was shown to be detrimental to the parasitic disease invasion and transmission, which implies that the potential of using artificial disturbances as a disease-control agency in biological conservation and management. Two components of the spatial heterogeneity (the amount and spatial autocorrelation of the lost habitat) formed a trade-off in determining the host–parasite dynamics. An extremely high degree of habitat loss was, counter-intuitively, harmful to the host. These results enrich our understanding of eco-epidemiological, host–parasite systems, and suggest the possibility of using the spatial arrangement of habitat patches as a conservation tool for guarding focal species against parasitic infection and transmission.

Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System

Epidemic transmission is one of the critical density-dependent mechanisms that affect species viability and dynamics. In a predator-prey system, epidemic transmission can strongly affect the success probability of hunting, especially for social animals. Predators, therefore, will suffer from the positive density-dependence, i.e., Allee effect, due to epidemic transmission in the population. The rate of species contacting the epidemic, especially for those endangered or invasive, has largely increased due to the habitat destruction caused by anthropogenic disturbance. Using ordinary differential equations and cellular automata, we here explored the epidemic transmission in a predator-prey system. Results show that a moderate Allee effect will destabilize the dynamics, but it is not true for the extreme Allee effect (weak or strong). The predator-prey dynamics amazingly stabilize by the extreme Allee effect. Predators suffer the most from the epidemic disease at moderate transmission probability. Counter-intuitively, habitat destruction will benefit the control of the epidemic disease. The demographic stochasticity dramatically influences the spatial distribution of the system. The spatial distribution changes from oil-bubble-like (due to local interaction) to aggregated spatially scattered points (due to local interaction and demographic stochasticity). It indicates the possibility of using human disturbance in habitat as a potential epidemic-control method in conservation.

Evolution of cooperation in patchy habitat under patch decay and isolation

The spatial version of Prisoner’s Dilemma (PD) is studied, which incorporates habitat decay through change in the mortality parameter and habitat isolation through change in the colonization coefficient. We found four kinds of evolutionary results, which are affected profoundly by the elements of the payoff matrix and the ratio of the colonization coefficient to the mortality parameter: population extinction, a pure cooperator population, coexistence of cooperators and defectors, and a pure defector population. First, the parameter region of cooperation (pure cooperator and coexistence region) shrinks with an increase in the cooperative cost, and that of defection extends. The increase in cooperative reward makes the cooperative region extend and the defector region become small. Second, the cooperative reward can compensate for the extinction risk due to habitat destruction and allow a population to survive even if the colonization coefficient is smaller than the mortality parameter. Third, although habitat destruction (including decay and isolation) increase the extinction risk of a population, moderate external power can push the evolution of cooperation ahead of population extinction, and even make a completely cooperative world come into being. Finally, for certain values of elements of the payoff matrix, the population suffering habitat destruction can maintain a stable population size by regulating the frequencies of cooperators and defectors. This implies that the multi-behavior strategy within a population may be a mechanism to defend against the influences of a changing environment.