Stable Positive Equilibrium Research Articles

Stabilizing control of ratio-dependent predator–prey models

In this paper, the control of a ratio-dependent predator–prey system is considered. This system, in its equivalent nondimensional form, is subjected to an on–off control, also known as a threshold policy. The control, applied only to the predator, is of the “constant effort removal” type. It is shown that such a control can be designed by a suitable choice of so called virtual equilibrium points. The analysis of the global stability of a specific nondimensional model is carried out. It shows that, in the nondimensional model, the on–off control changes the original equilibrium points on the axes to a globally stable positive equilibrium point, attained through a sliding mode, thus avoiding extinction, and achieving coexistence of both species.

The effect of harvesting on a predator–prey system with stage structure

This paper studies the effect of harvesting and stability of a stage-structured predator–prey model with two life stages, immature and mature, and with a mature population of harvesting. We establish conditions for the existence of a globally asymptotically stable positive equilibrium and a threshold of harvesting for mature population.

Dynamic complexities of a Holling I predator–prey model concerning periodic biological and chemical control

We investigate the dynamic behaviors of a Holling I predator–prey model with impulsive effect concerning biological and chemical control strategy––periodic releasing natural enemies and spraying pesticide at different fixed time. By using Floquet theorem and small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. The condition for the permanence of the system is given. It is shown that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Further, the effects of impulsive perturbations on the unforced continuous system is studied. We find that the system we considered has more complex dynamic behaviors and is dominated by periodic, quasi-periodic and chaotic solutions. We also find that our impulsive forced system may have different dynamic behaviors with different range of initial values, with which the solutions of the unforced system tend either to the inherent stable limit cycle or to a stable positive equilibrium.

A Mathematical Model of Competition for Two Essential Resources in the Unstirred Chemostat

A mathematical model of competition between two species for two growth-limiting, essential (complementary) resources in the unstirred chemostat is considered. The existence of a positive steady-state solution and some of its properties are established analytically. Techniques include the maximum principle, the fixed point index, and numerical simulations. The simulations also seem to indicate that there are regions in parameter space for which a globally stable positive equilibrium occurs and that there are other regions for which the model admits bistability and even multiple positive equilibria.

Intraspecific interference and consumer-resource dynamics

In this paper we first consider a two consumer-one resource modelwith one of the consumer species exhibits intraspecific feedinginterference but there is no interspecific competition between thetwo consumer species. We assume that one consumer species exhibitsHolling II functional response while the other consumer species exhibitsBeddington-DeAngelis functional response. Using dynamical systems theory,it is shown that the twoconsumer species can coexist upon the single limiting resource inthe sense of uniform persistence. Moreover, by constructing a Liapunov functionit is shown that the systemhas a globally stable positive equilibrium. Second, we consider a modelwith an arbitrary number of consumers and one single limiting resource.By employing practical persistence techniques, it is shown that multipleconsumer species can coexist upon a single resource as long as all consumersexhibit sufficiently strong conspecific interference, that is, each of themexhibits Beddington-DeAngelis functional response.

Optimal harvesting and stability for a two-species competitive system with stage structure

In this paper, we consider a stage-structured competitive population model with two life stages, immature and mature, with a mature population of harvesting. We obtain conditions for the existence of a globally asymptotically stable positive equilibrium and a threshold of harvesting for the mature population. The optimal harvesting of the mature population is also considered.

A discrete analogue of an integrodifferential equation

A discrete version of the integrodifferential equation dx(t) dt =x(t) a−x(t)−b ∫ −∞ t k(t−s)x(s)ds is formulated. It is shown that the discrete equation has a globally asymptotically stable positive equilibrium under the same conditions with which the positive equilibrium of the integrodifferential equation is globally asymptotically stable. The integrodifferential equation has applications in population dynamics.

Global stability in a nonlinear difference equation

Consider the nonlinear difference equation where By using Liapunov's method, we establish some sufficient conditions for Eq. (0.1) to have a globally asymptotically stable positive equilibrium. Our results can be applied to some difference equations derived from mathematical biology.

Computer aided proof for the global stability of Lotka-Volterra systems

By modifying the mechanical method of determining LaSalle's invariant sets for Lotka-Volterra chain systems [1,2] and repeatedly using Wu's characteristic set method [3,4], it is proved that for a class of Lotka-Volterra loop systems, the locally asymptotically stable positive equilibrium point must be globally stable.

A mechanistic model of the adaptation of phytoplankton photosynthesis

The mechanistic model of the phytoplankton photosynthesis-light intensity relationship by Eilers and Peeters (1988.Ecol. Modelling 42, 199–215) is investigated mathematically. The model is based on the physiological idealization of transition probabilities between states of the photosynthetic factories,PSF. The model was found to have under constant light condition a globally stable unique positive equilibrium, while under periodically varying light (e.g. daily periodicity) there exists a unique globally asymptotically stable periodic solution. Based on this, the adaptation to a change of light intensity is defined as a process by which the state ofPSF converges to an equilibrium. Assuming that phytoplankton regulates its photosynthetic production rate with a certain strategy which maximizes production, two such possible strategies were examined. Both the instantaneous and the integral maximal photosynthetic production were shown to have the same result. With realistic qualitative assumptions of the shape of the dependence of the four model parameters on the light intensity to which phytoplankton is adapted, the numerical values of parameters under both constant and periodically varying conditions are determined by applying Pontryagin's maximum principle.

The Owl’s Odyssey. A Continuous Model for the Dispersal of Territorial Species

In this paper, a composite model is developed that consists of a continuous model for dispersal set within a difference equation model for the life history of territorial species. Two dispersal models are considered: one that assumes that suitable habitat is uniformly or randomly distributed throughout the range of the population, and one that assumes that home ranges are concentrated in clusters of suitable habitat. These models explicitly consider the cost of dispersal by including ongoing rates of mortality due to predation and starvation while individuals search for a territory. The cluster model also differentiates between mortality within and outside of clusters. An analysis of the difference equation model demonstrates a threshold for density of suitable habitat below which the population must decrease to extinction, and above which the population tends monotonically to a stable positive equilibrium size. In addition, it is established that the equilibrium size of the population can be increased by...

A size-structured model for cannibalism

A size-structured model for the dynamics of a cannibalistic population is derived under the assumption that cannibals (successfully) attack only smaller bodied victims, as is generally the case in the biological world. In addition to the resulting size-dependent death rate, the model incorporates the positive feedback mechanism resulting from the added resource energy obtained by the cannibal from the consumption of the victim. From the nonlinear partial integro-differential equation model, it is shown how to obtain a complete analysis of the global dynamics of the total population biomass. This analysis yields many dynamical features that have been attributed to cannibalism in the literature, including density self-regulation, a “life-boat strategy” phenomenon by which a population avoids extinction by practicing cannibalism under circumstances when it would otherwise go extinct, and multiple stable positive equilibrium states and hysteresis.

Liénard-type equations and the epidemiology of malaria

Abstract Lienard's equation and Liapunov's direct method are applied in the dynamics of malarial transmission to guarantee a globally asymptotically stable positive equilibrium or a stable persistence of infection in terms of the rate of biting per female mosquito on unifected hosts.

A time-delay model of single-species growth with stage structure

A single-species growth model with stage structure consisting of immature and mature stages is developed using a discrete time delay. It is shown that under suitable hypotheses there exists a globally asymptotically stable positive equilibrium. Questions concerning oscillation and nonoscillation of solutions are addressed analytically and numerically. The effect of the delay on the populations at equilibrium is also considered.

A general minimum principle for competing populations: Some ecological and evolutionary consequences

The principle introduced by MacArthur (1969. Proc. Natl. Acad. Sci. U.S.A., 64, 1369–1375) that the stable positive equilibrium of an exploitative competition system minimizes the mean squared deviation between available and utilized resources is generalized. It is shown that under less stringent assumptions, the global attractor of a competition system consists of species assortments that minimize a function of consumers' abundances, called inefficient energy utilization. If the minimum is unique (whether or not lying on the boundary of the state space) the global attractor reduces to a globally stable equilibrium. The inefficient energy utilization is the sum of two terms: one, called unutilized productivity, is the squared difference between maximum resource production and consumers' consumption, while the second, called basal energy consumption, is the caloric income to a community necessary for maintaining it at a steady state. The minimum principle is then used to solve some ecological and evolutionary problems. It is demonstrated that if an inverse correlation is assumed between consumers' efficiency and width of the trophic niche, evolution in undisturbed environments leads to assortments of tightly packed specialists.

Predator survival versus extinction as a function of dispersal in a predator–prey model with patchy environment

This paper is concerned with a model of a predator–prey system, where both populations disperse among n patches forming their habitat. Criteria are given tor both survival and extinction of the predator population. In case the predator survives, conditions are derived which guarantee a globally asymptotically stable positive equilibrium

Equilibria in systems of interacting structured populations.

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k-1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.

Neighborhood Models of Plant Population Dynamics. 4. Single-Species and Multispecies Models of Annuals with Dormant Seeds

Neighborhood population-dynamic models are developed for single-species and multispecies communities of annual plants with seeds that may lie dormant through one or more growing seasons. Dormancy-induced changes in lifetime germination successes affect the equilibrium densities of plants and the local stability of equilibria. Dormancy-induced population-dynamic time lags, in contrast, do not affect equilibrium densities of post-germination plants or the ability of an invader to successfully colonize a multispecies community. In single-species populations, time lags are never destabilizing but may be stabilizing. Specifically, some strategies of dormancy (a way to partition lifetime germination success among the various age classes of seeds) cause an otherwise oscillatory monoculture (eigenvalue less than negative one) to have a locally stable positive equilibrium. The maximum distance over which a plant of one species can interfere with a plant of another species is termed the neighborhood radius. The pop...

Global stability in time-delayed single-species dynamics.

Criteria are established for three classes of models of single-species dynamics with a single discrete delay to have a globally asymptotically stable positive equilibrium independent of the length of delay.

Global stability in time-delayed single-species dynamics

Criteria are established for three classes of models of single-species dynamics with a single discrete delay to have a globally asymptotically stable positive equilibrium independent of the length of delay.