214 research outputs found
On a predator prey model with nonlinear harvesting and distributed delay
A predator prey model with nonlinear harvesting (Holling type-II) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some
numerical simulations.Ministerio de EconomÃa y CompetitividadFondo Europeo de Desarrollo RegionalConsejerÃa de Innovación, Ciencia y Empresa (Junta de AndalucÃa
Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model
In the natural world, there are many insect species whose individual members
have a life history that takes them through two stages, immature and mature.
Moreover, the rates of survival, development, and reproduction almost always
depend on age, size, or development stage. Keeping this in mind, in this paper,
a three species crop-pest-natural enemy food chain model with two stages for
natural enemies is investigated. Using characteristic equations, a set of
sufficient conditions for local asymptotic stability of all the feasible
equilibria is obtained. Moreover, using approach as in (Beretta and Kuang,
2002), the possibility of the existence of a Hopf bifurcation for the interior
equilibrium with respect to maturation delay is explored, which shows that the
maturation delay plays an important role in the dynamical behavior of three
species system. Also obtain some threshold values of maturation delay for the
stability-switching of the particular system. In succession, using the normal
form theory and center manifold argument, we derive the explicit formulas which
determine the stability and direction of bifurcating periodic solutions.
Finally, a numerical simulation for supporting the theoretical analysis is
given.Comment: 28 pages, 9 figure
Hopf bifurcation and optimal control in a diffusive predator-prey system with time delay and prey harvesting
In this paper, we investigated the dynamics of a diffusive delayed predator-prey system with Holling type II functional response and nozero constant prey harvesting on no-flux boundary condition. At first, we obtain the existence and the stability of the equilibria by analyzing the distribution of the roots of associated characteristic equation. Using the time delay as the bifurcation parameter and the harvesting term as the control parameter, we get the existence and the stability of Hopf bifurcation at the positive constant steady state. Applying the normal form theory and the center manifold argument for partial functional differential equations, we derive an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, an optimal control problem has been considered
Controllability of an eco-epidemiological system with disease transmission delay: A theoretical study
This paper deals with the qualitative analysis of a disease transmission delay induced prey preda-tor system in which disease spreads among the predator species only. The growth of the preda-tors’ susceptible and infected subpopulations is assumed as modified Leslie–Gower type. Suffi-cient conditions for the persistence, permanence, existence and stability of equilibrium points are obtained. Global asymptotic stability of the system is investigated around the coexisting equilib-rium using a geometric approach. The existence of Hopf bifurcation phenomenon is also exam-ined with respect to some important parameters of the system. The criterion for disease a trans-mission delay the induced Hopf bifurcation phenomenon is obtained and subsequently, we use a normal form method and the center manifold theorem to examine the nature of the Hopf bifurca-tion. It is clearly observed that competition among predators can drive the system to a stable from an unstable state. Also the infection and competition among predator population enhance the availability of prey for harvesting when their values are high. Finally, some numerical simu-lations are carried out to illustrate the analytical results
Hopf Bifurcation in a Modified Leslie-Gower Two Preys One Predator Model and Holling Type II Functional Response with Harvesting and Time-Delay
In this paper, a modified Leslie-Gower two preys one predator model and Holling type II functional response with harvesting and time-delay were discussed. Model analysis is carried out by determining fixed points, then analyzing the stability of the fixed points and discussing the existence of the Hopf bifurcation. In some conditions that occur in nature indicate the occurrence of hunting of prey and predator species by humans. Therefore, this model is modified by adding the assumption that prey and predators are being harvested. Another modification given to the model is the use of time delays.The delay time term is for taking into account the case that the members of the predator species need time from birth to predation for being active predators. The first case is a model without time delay, it is obtained that 3 fixed points are unstable and 7 fixed points are stable. One of them is the interior fixed point tested with the Routh-Hurwitz criteria. The second case is a model with a delay time, the critical delay value is obained. Hopf bifurcation occurs when the delay time value is equal to the critical delay value and also fulfills the transversality condition. Observations on the model simulation are carried out by varying the value of the delay time. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of the delay time given is less than the critical value of the delay, the controlled system solution goes to a balanced state. Then when the delay time value is greater than the critical delay value, the system solution continues to fluctuate causing an unstable system condition
Global stability and bifurcation analysis of a delayed predator-prey system with prey immigration
A delayed predator-prey system with a constant rate immigration is considered. Local and global stability of the equilibria are studied, a fixed point bifurcation appears near the boundary equilibrium and Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values. We also show the existence of the global Hopf bifurcation, and the properties of the fixed point bifurcation and the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem
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