Stochastic periodic solution for a perturbed non-autonomous predator–prey model with generalized nonlinear harvesting and impulses

Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

“…e proof is similar to that of eorem 3.1 in [20] and is omitted. Lemma 3 reveals the equivalence of (2) and (15), and hence, we will consider (15) later.…”

“…Assume that the product equals unity if the number of factors is zero. Obviously, A 1 (t)and A 2 (t) are both T-periodic (for details, see [20]).…”

“…Proof. Firstly, the proof of stochastic persistence in probability is motivated by [20]. Let (u 1 (t, u 1 (0)), u 2 (t, u 2 (0))) be the solution of (15) with initial data (u 1 (0), u 2 (0)) ∈ R 2 + , p(t, u(0), dy) be the transition probability of u(t, u(0)), and P(t, u(0), Γ) be the probability of events u(t, u(0)) ∈ Γ. e stochastic boundedness of u(t, u(0)) implies there exists a compact subset K ⊆ R 2 + such that P(t, ϕ, K) ≥ 1 − ε for any given ε > 0, that is, p(t, u(0), dy): t ≥ 0 is tight.…”

“…In natural world, due to individual life cycle and seasonal variation, the carrying capacity of species, birth rate, and other parameters always present periodic changes for population systems [16][17][18]. For the determinate biological system, the existence of periodic solution is a very important dynamical behavior [10,[19][20][21][22]. Similarly, for stochastic system, it is very interesting to study the existence of stochastic periodic solution (periodic Markovian process).…”

Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator‐Prey Competitive System with Lévy Jumps

“…However, the above two classes of models can not describe some natural phenomena completely and it is believed that models with three or more species can explain the dynamical behaviors of the population accurately [5][6][7][8]. Predator-prey models are arguably known as the most fundamental building blocks of the biosystems and ecosystems as the biomasses are grown out of their resource, which have been widely investigated [9][10][11][12][13][14][15][16][17]. As we all know, after a predator catches and feeds on a prey, the number of the predators will not increase at once, which needs the processes of digestion and absorption.…”