DOUBLE ALLEE EFFECTS ON PREY IN A MODIFIED ROSENZWEIG-MACARTHUR PREDATOR-PREY MODEL

Abstract

In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.In this work, a modified Rosenzweig-MacArthur predator-prey model is analyzed, which is a particular Gause type model, considering two Allee effect affecting the prey population. This phenomenon may be expressed by different mathematical expressions; with the form here used, the existence of one limit cycle surrounding a positive equilibrium point is proved. Conditions to the existence of equilibrium points and their local stability are established; moreover, the existence of a separatrix curve dividing the behavior of trajectories which can have different ?-limit sets. Some simulations reinforced our results are given and the ecological consequences are discussed.