Stability Of Predator-prey Model Research Articles

Optimal harvesting and stability of predator-prey model with holling type II predation respon function and stage-structure for predator

This article develops mathematical models with structural stages from predators, immature and mature predators. The predation function of mature predators follows the Holling II response function. We assume that the immature predator population has the economic value, therefor the harvesting function is included in this model. In this model an analysis of the equilibrium point and stability of the interior equilibrium point is carried out. Analysis of the stability of the interior equilibrium points is done by linearization method and pay attention to the eigenvalues of the characteristics of the Jacobi matrix obtained. Analysis of equilibrium point stability is carried out before and after harvesting. The result is obtained by each of the three equilibrium points. At the equilibrium point of the interior with stable harvesting a local maximum profit analysis is obtained from the exploitation business. Based on the results of the analysis, it is obtained the value of harvesting business which provides a stable equilibrium point and maximum profit.

Optimal harvesting and stability of predator prey model with Monod-Haldane predation response function and stage structure for predator

This article deals with growth rate predator prey population model with stage structure for predator. The predator population is divided into two stages, mature and immature predator. Predation functional response in the model follows the Monod-Haldane type. Under consideration that the population is a valuable stock, the mature predator and prey population are then harvested with constant effort. This model is then analysed by found the equilibrium point and stabilities. There exists four equilibrium points and stability analysed is focused only for interior point. The stable interior point is related to the maximum profit problem. From the analyses we found that there exists a condition for the effort of harvesting so that the interior point is still stable and also obtained maximum profit.

Multiple limit cycles and global stability in predator-prey model

A predator-prey system is investigated for the global stability and existences of limit cycle and at least two cycles by using qualitative analysis and the idea of Poincare-Bendixson Theoy.

Short Term Refuge Use and Stability of Predator-Prey Models

An individual-level behavioural model representing prey becoming temporarily less susceptible to attack in response to detection of predators is presented. This model is used to obtain a functional response for the predator which encapsulates a prey-hiding effect A density-dependent term is induced in the functional response which is structurally identical to that derived by Ruxton et al. (1992) using a model representing mutual interference between predators. The effect of the hiding response on the vital rates of the prey population is also derived from the behavioural model. Two scenarios are explored: one where hiding prevents prey from reproducing and another where hiding increases mortality from non-predation sources. These effects (and the functional response) are incorporated into two simple population-level predator-prey model. Prey hiding has a strong stabilising effect against prey escape cycles in both models.

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).