Permanence, partial survival, extinction, and global attractivity of a nonautonomous harvesting Lotka–Volterra commensalism model incorporating partial closure for the populations

Abstract

We propose and study a nonautonomous harvesting Lotka-Volterra commensalism model incorporating partial closure for the populations. By using the differential inequality theory we obtain sufficient conditions that ensure the extinction, partial survival, and permanence of the system. By applying the fluctuation lemma we establish sufficient conditions that ensure the extinction of one of the components and the stability of the the other one. For the permanent case, by constructing a suitable Lyapunov function we obtain some sufficient conditions for the globally attractivity of the positive solution of the system. Examples, together with their numeric simulations, show the feasibility of the main results. To ensure the stable coexistence of the two species, the harvesting area should be carefully restricted.

Note on the stability property of the boundary equilibrium of a May cooperative system with strong and weak cooperative partners

“…During the last decade, many scholars ( [2][3][4][5][6][7][8][9][10][11][12][13][14][16][17][18][19][20][21][22][23][24][25]) investigated the dynamic behaviors of the mutualism and commensalism model. Some substantial progress has been made on the stability, permanence and extinction of the mutualism model.…”

Dynamic behaviors of a stage-structured commensalism system

“…On the other hand, many scholars investigated the dynamic behaviors of the stagestructured species, see [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45], and the references cited therein. In constructing a stagestructured model, two different ideas were applied.…”

“…(1.8) They showed that system (1.8) admits a unique positive equilibrium which is globally asymptotically stable. Many scholars [26][27][28][29][30][31][32][33][34][35][36][37][38][39] used the idea of Aiello and Freedman to establish stage-structured ecological models. For example, Chen et al [28] studied the persistence property of the following stage-structured predator-prey model:…”

Dynamic Behavior of a Commensalism Model with Nonmonotonic Functional Response and Density‐Dependent Birth Rates

“…An example of it is that remora are specially adapted to attach themselves to larger fish that provide locomotion and food. In the last decades, commensalism has attracted the attention of many researchers ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). Complicated dynamics have been found in the study.…”

“…For more details, see [6][7][8]. Combining this with (1), we propose the following commensalism model:…”