Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy

Complex dynamics of two prey–predator harvesting models with prey refuge and interval‐valued imprecise parameters

“…In this subsection, we present concepts and results of impulsive semi-dynamical system (ISDS) here for the convenience of readers [38,55]. Let us consider a planar impulsive model with threshold { dx dt = P(x, 𝑦), d𝑦 dt = Q(x, 𝑦) 𝜙(x, 𝑦) ≠ 0, Δx = I 1 (x, 𝑦), Δ𝑦 = I 2 (x, 𝑦) 𝜙(x, 𝑦) = 0, (A1) where P, Q, 𝛼, b and 𝜙 are differentiable with respect to x and 𝑦. Denote W as the domain of solutions, 𝜋 = (𝜋 1 , 𝜋 2 ) ∶ W ×R → W characterize the solution map of the corresponding continuous system, and define  ≜ {(x, 𝑦) ∈ W|𝜙(x, 𝑦) = 0}, I = (I 1 , I 2 ):  →  = I().…”

“… many times due to the pulse action, the set of the time is denoted by [38,55]). z(t) = (x(t), ỹ(t)) ′ with z0 ∈  is said to be periodic if ∃n ⩾ 1 such that…”

“…Therefore, scientific and rational resource management plan is the guarantee of fishery sustainable development. Fishing activity is a typical human activity, which can be carried out in continuous forms [31][32][33][34][35] or discontinuous forms [36][37][38][39][40]. In continuous form of harvesting, the modeling is more flexible and can be directly implemented by adding a term representing harvest to the predator-prey model.…”

“…Thus, impulsive differential equations (IDEs) provide a natural and effective description in modelling such processes, such as integrated pest management [41][42][43], prey-predator system [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], competitive system [60][61][62]. In prey-predator system, the human interventions can be divided into prey-dependent [37,41-43, 47, 48, 51-57], predator-dependent [36,37] and ratio-dependent [63] and prey-predator hybrid-dependent [38,39,64,65]. In natural systems, predators and prey are mixed, so it is impossible to determine the exact number of the two species, but their proportions are usually kept constant.…”

“…In natural systems, predators and prey are mixed, so it is impossible to determine the exact number of the two species, but their proportions are usually kept constant. Based on this consideration, a weight capture strategy was introduced into a fishery model [32,38], where fishing activities are allowed when the weighted sum of the populations of the two species reaches a threshold. Combining practical significance of refuge effect and imprecise biological parameters in predator-prey model, in the current work, we present a prey-predator model with prey refuge and two types of harvesting strategies in an imprecise parameter environment, and discuss the effects of refuge factors and imprecise indicators on system dynamics and the complex dynamics induced by the harvesting activities.…”