In this paper, a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps is studied. Firstly, it is proved that the model has a unique global solution. Secondly, sufficient conditions for weak persistence in the mean and extinction of prey and predator populations are established. Finally, sufficient conditions for the existence and uniqueness of ergodic stationary distribution are established. Moreover, several numerical simulations are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]
This paper presents an optimal control of intervention strategies for the menace of Banditry taking into account media campaign against Banditry u2(t) rehabilitation of Bandits detainees u5(t) and use of military force against Banditry u6(t) as control strategies. The Banditry free equilibrium, Banditry present equilibrium and the basic reproduction number of Banditry R0B were obtained. The stability analysis results suggest that the Banditry free equilibrium is locally asymptotically stable when R0B>1 and otherwise when R0B>1. The Boko Haram presence equilibrium is globally asymptotically stable when R0B>1 and unstable if R0B>1. We used the three control strategies and updated the Banditry menace model. The optimal control issue was resolved using Pontryagin's Maximum Principle (PMP). It was discovered that there is a significant decrease in the population of Bandits and increase in the number of rehabilitated Bandits and detained Bandits when the control measures are implemented compared to the case without control. We solved the optimality control using a forward-backward sweep strategy implemented in MATLAB for numerical simulation. Additionally, we saw that the number of people detained fluctuates as the number of people receiving rehabilitation rises. We argue that in order to lessen or completely erase the menace caused by Bandits in society, the government should fund media campaigns and rehabilitation initiatives. [ABSTRACT FROM AUTHOR]
A single-species population model exhibiting a symmetric slow variation for the carrying capacity and intrinsic growth rate is evaluated explicitly. However, it is unrealistic to eliminate the possibility of a clear separation in the evolution of the biotic environmental elements; thus, this paper considers the situation where these elements have a hierarchical variation on the time scales. Accordingly, two particular situations are recognized, which are the carrying capacity varies faster than the growth rate and vice versa. Applying the multi-time scaling technique in such a system provides a small parameter, which leads us to construct analytical approximate expressions for the population behavior, using the perturbation approach. Such approximations display very good agreement with the numerical simulations. [ABSTRACT FROM AUTHOR]