A stage-structured predator–prey model with distributed maturation delay and harvesting

Abstract

A stage-structured predator-prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium. ARTICLE HISTORY

The Dynamical Analysis of a Delayed Prey-Predator Model with a Refuge-Stage Structure Prey Population

“…In general, the time delays in mathematical models of population dynamics are due to maturation time, gestation time, capturing time or some other reasons and since in most applications of delay differential equations in biology, the need for incorporating time delays is often due to the existence of some stage structures. So, some works of stage-structure preypredator models with time delay have been provided in the literatures [17][18][19][20][21][22][23][24][25][26]. Bandyopadhyaya and Banerjee in [20], Yuanyuan and Changming in [21], and Wang et al in [22] proposed three mathematical models of stage-structure preypredator involving time delay for gestation, which based on the fact that the reproduction of predator will not be instantaneous after eating the prey but mediated by some time delay needed for gestation of predator, in their models, they supposed that the predator feeds on the immature prey only or mature prey only, and ignored the predation of the other prey.…”

Implication of age-structure on the dynamics of Lotka Volterra equations

“…(t, a) + ∂x ∂a (t, a) = −µ(a)x(t, a) − y(t)γ(a)x(t, a), dy dt (t) = αy(t) ∞ 0 γ(a)x(t, a)da − δy(t), x(t, 0) = ∞ 0 β(a)x(t, a)da ∀t ≥ 0, x(0, a) = x 0 (a) ∀a ≥ 0 and y(0) = y 0 , (1) with x(t, a) and y(t) that are respectively the density of preys at age a ≥ 0 and time t ≥ 0 and the density of predators at time t where:…”

“…These results are obtained by considering model (1) as a semilinear abstract Cauchy Problem and using semigroup theory. Section 3 is devoted to the main results of the article: one can exhibits an explicit formulation of a threshold for the extinction of the total population, by performing a stability analysis of equilibria of Problem (1). The use of spectral theory for the differential operator of the PDE problem, and some compactness properties of the nonlinear part of the problem are the main arguments to achieve that goal.…”

Stability analysis and optimal impulsive harvesting for a delayed stage-structured self dependent two compartment commercial fishery model

“…[22] developed a stagestructured predator-prey model by using the equations similar to the model equations in Equation ( 1) with stage-structuring in both prey and predator and established the permanence of the system. Many other (see [23][24][25]) applied the equations similar to the equations in model Equation ( 1) to establish the prey-predator model, competition models and obtained many other interesting results. But during literature survey, we could not find impulsive fish harvesting model using the equations developed by Aiello and Freedman as given in Equation (1).…”