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This paper focuses on asymptotic mean-square boundedness of several numerical methods applied to a class of stochastic age-dependent population equations with Poisson jumps. The conditions under which the underlying systems are asymptotic mean-square boundedness are considered. It is shown that the asymptotic mean-square boundedness is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce asymptotic mean-square boundedness under a stepsize constraint. The results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of asymptotic mean-square boundedness. Finally, an example is given for illustration. [ABSTRACT FROM AUTHOR]

Yongzhen et al. (2010) describe a mathematical model of a scenario where a plant population is imported to a pest–predator system with an infected pest. Thus a plant–pest–predator eco-epidemiological model disturbed by an impulsive effect is proposed. First of all, the stability conditions of the susceptible pest-eradication periodic solution for eradicating the susceptible pest are investigated. Compared with the results in (Yongzhen et al., 2010), the presence of the plant population increases the cost of natural enemies as well as the demand for insecticide. In addition, we study the effect of the death rate of the infected pest on pest control in terms of evolution of virulence and the basic reproductive number. Results show that larger mortalities of the infected pest will lead to the frustrated invasion or the instability of susceptible pest-eradication periodic solutions. Next, we focus on the four kinds of optimal impulsive control strategies, biological control, chemical control, and integrated control with fixed period or variable period, to maximize the yields of plants at the terminal time with minimum efforts. All the optimal control problems are solved via a time scaling technique and a gradient-based optimization method. Our results show that two parameters, the amount of sprayed infective pest and the kill fraction of the susceptible pest, play a key role in improving the yield of the plants. In addition, for the four kinds of control strategies, our results also show that biological control is more effective than chemical control to achieve an optimal solution, and the last two strategies can produce higher yields than the first two control strategies. [ABSTRACT FROM AUTHOR]

Abstract: Taking into account periodic impulsive biological and chemical control for pest management at different fixed moment, a three species prey–predator system with Holling type II functional response was investigated. By using Floquet’s theory and the small amplitude perturbation method, it was obtained that there exists an asymptotically stable preys-eradication periodic solution when the impulsive period is less than some critical minimum value (or the release amount of the predator is larger than some critical maximum value), and the system is permanent under the conditions that both the insecticidal effect and impulsive period are grater than some critical maximum values. Furthermore, it is obtained that IPM is more effective than any single one after comparison. Finally, numerical simulations are carried on to show the complex dynamic behavior of system. [Copyright &y& Elsevier]

The aim of this paper is to construct a numerical method preserving positivity for stochastic age-dependent population equations. We use the balanced implicit numerical techniques to maintain the nonnegative path of the exact solution. It is proved that the Balanced Implicit Method (BIM) preserves positivity and converges with strong order 1/2 under given conditions. Finally, two examples are simulated to verify the positivity and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Considering many species can release toxic substances to protect themselves against predators, a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated. For this system, we give a complete and rigorous analysis of the global dynamics with the boundedness, globally asymptotical stability, transcritical bifurcation, Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions. Meanwhile, we consider the effects of toxins produced by the prey on the dynamic behavior. The consequence of the global stability of the coexistence equilibrium is that the toxin’s intrinsic characteristic will not change the stability of the system irreversibly. Our results show that the toxin-produced by the prey (phytoplankton) may be used as a bio-control agent for the Harmful Algal Bloom problems. [ABSTRACT FROM AUTHOR]

In this paper, a new split-step θ (SS θ ) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS θ method for stochastic age-dependent population equations with Poisson jumps. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory, the results show that the SS θ method has better accuracy compared to the Euler method. [ABSTRACT FROM AUTHOR]

Abstract: In compared with specialist predators which feed almost exclusively on a specific species of prey, generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population, and the generalist predators has itself growth function which be extended a well-known logistic growth term. We develop a generalist predator–prey model with diffusion and study the effect of harvesting and delay under Neumann conditions. The stability of the equilibria is firstly investigated, and the existence of traveling wave solutions is then established by constructing a pair of upper–lower solutions and using the cross iteration method and Schauder’s fixed point theorem. [Copyright &y& Elsevier]