Global stability in two species interactions

Asymptotic properties of the Lotka-Volterra competition and mutualism model under stochastic perturbations

Stochastically perturbed models, where the white noise type stochastic perturbations are proportional to the current system state, the most realistically describe real-life biosystems. However, such models essentially have no equilibrium states apart from one at the origin. This feature makes analysis of such models extremely difficult. Probably, the best result that can be found for such models is finding of accurate estimations of a region in the model phase space that serves as an attractor for model trajectories. In this paper, we consider a classical stochastically perturbed Lotka-Volterra model of competing or symbiotic populations, where the white noise type perturbations are proportional to the current system state. Using the direct Lyapunov method in a combination with a recently developed technique, we establish global asymptotic properties of this model. In order to do this, we, firstly, construct a Lyapunov function that is applicable to the both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method allows to find estimates for this region. We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization ('stabilization by noise'). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (that for this model occurs via the process that is known as the 'stochastic stabilization', or the 'stabilization by noise') are found applying the Khasminskii-type Lyapunov functions.

A survey on Lyapunov functions for epidemic compartmental models

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.

A remark on "COVID-19: Perturbation dynamics resulting chaos to stable with seasonality transmission" [Chaos, Solitons and Fractals 145 (2021) 110772]

In Batabyal (2021)[2], introducing an extension of the well-known susceptible-exposed-infected-recovered (SEIR) model with seasonality transmission of SARS-CoV-2, the author has derived and discussed various analytical and numerical results. Careful scrutiny of the said article brings about some genuine issues pertaining to the model formulation, analysis and numerical studies carried out in Batabyal (2021)[2]. Given the present pandemic and the havoc it has been causing throughout the world, and the responsibility of giving out rightful information/results backed by scientific proofs, there is a pressing need to address issues of such kind right away.

On the dynamics of a class of perturbed cyclic Lotka-Volterra systems

We consider a special class of Lotka–Volterra systems where the associated interaction matrix is cyclic, but asymmetric, with a perturbation term on each row. After some discussion of the dynamics under a general setting, we focus our attention on 3D systems for a more detailed study. We derive sufficient conditions for the existence and stability of the nontrivial interior equilibrium. We also show that Hopf bifurcation occurs when the size of the perturbation is large. Such analysis can be similarly extended to higher dimensional systems, and we mention some results in 4D case.

Global stability of infectious disease models with contact rate as a function of prevalence index

In this paper, we consider a SEIR epidemiological model with information--related changes in contact patterns. One of the main features of the model is that it includes an information variable, a negative feedback on the behavior of susceptible subjects, and a function that describes the role played by the infectious size in the information dynamics. Here we focus in the case of delayed information. By using suitable assumptions, we analyze the global stability of the endemic equilibrium point and disease--free equilibrium point. Our approach is applicable to global stability of the endemic equilibrium of the previously defined SIR and SIS models with feedback on behavior of susceptible subjects.

The potential and flux landscape theory of ecology

The species in ecosystems are mutually interacting and self sustainable stable for a certain period. Stability and dynamics are crucial for understanding the structure and the function of ecosystems. We developed a potential and flux landscape theory of ecosystems to address these issues. We show that the driving force of the ecological dynamics can be decomposed to the gradient of the potential landscape and the curl probability flux measuring the degree of the breaking down of the detailed balance (due to in or out flow of the energy to the ecosystems). We found that the underlying intrinsic potential landscape is a global Lyapunov function monotonically going down in time and the topology of the landscape provides a quantitative measure for the global stability of the ecosystems. We also quantified the intrinsic energy, the entropy, the free energy and constructed the non-equilibrium thermodynamics for the ecosystems. We studied several typical and important ecological systems: the predation, competition, mutualism and a realistic lynx-snowshoe hare model. Single attractor, multiple attractors and limit cycle attractors emerge from these studies. We studied the stability and robustness of the ecosystems against the perturbations in parameters and the environmental fluctuations. We also found that the kinetic paths between the multiple attractors do not follow the gradient paths of the underlying landscape and are irreversible because of the non-zero flux. This theory provides a novel way for exploring the global stability, function and the robustness of ecosystems.

Chaotic dynamics of a three species prey-predator competition model with bionomic harvesting due to delayed environmental noise as external driving force

We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.

A MODEL FOR TWO SPECIES WITH STAGE STRUCTURE AND FEEDBACK CONTROL

This paper studies a periodic coefficients predator-prey delay system with mixed functional response, in which the prey has a history that takes them through two stages, immature and mature. Also, the total toxic action on the predator population expressed by an integral term is considered in our system. Furthermore, the feedback control is considered in our system. Sufficient conditions which guarantee the permanence and extinction of the system are obtained. Finally, we give a brief discussion of our results. From a biological point of view, our results can be used to help protect beneficial animals.

OUTPUT FEEDBACK CONTROL FOR AN EXPLOITED STRUCTURED MODEL OF A FISHING PROBLEM

In this paper, the problem of global state regulation via output feedback is investigated to study a structured fishing model, in order to stabilize its states around a non-trivial equilibrium. In our case, the two stages of the juvenile and adults ages of fish population are considered. In order to apply the tools of automatic control to our model, the fishing effort is used as a control term, the age classes as a states and the quantity of captured fish per unit of effort as a measured output. A Lyapunov function is adapted to study the stability and stabilization of the studied system around the non-trivial steady states. Numerical example demonstrates the effectiveness and the convergence of the states to the equilibrium.

Interspecific interactions between Bemisia tabaci biotype B and Trialeurodes vaporariorum (Hemiptera: Aleyrodidae)

Bemisia tabaci (Gennadius) biotype B and Trialeurodes vaporariorum (Westwood) are invasive whitefly species that often co-occur on greenhouse-grown vegetables in northern China. Although B. tabaci biotype B has been present in China for a relatively short period of time, it has become dominant over T. vaporariorum. We studied the interspecific competitive interactions between the two species in single or mixed cultures at 24 ± 1 °C, 40 ± 5% RH, and L14:D10 h photoperiod. Female longevity on tomato was not significantly different between species, but B. tabaci reproduced 4.3 to 4.9 fold more progeny. The ratio of female to male progeny in both instances was greater for B. tabaci. When cultured on tomato, cotton, and tobacco, B. tabaci developed 0.8, 3.3, and 4.7 d earlier in single culture, and 1.8, 3.9, and 4.3 d earlier in mixed culture. B. tabaci displaced T. vaporariorum in four, five and six generations when the initial ratios of B. tabaci to T. vaporariorum were 15:15, 20:10, or 10:20 on tomato. Populations of B. tabaci were 2.3 fold higher than that of T. vaporariorum on tomato plants for seven consecutive generations in single culture. B. tabaci performed better in development, survival, fecundity, and female ratio. We conclude that B. tabaci could displace T. vaporariorum in as short as four generations in a controlled greenhouse environment when they start at equal proportions. Warmer greenhouse conditions and an increase in total greenhouse area could be contributing factors in the recent dominance of B. tabaci.

Two models of interfering predators in impulsive biological control

In this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters.