Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey

Stationary distribution and global stability of stochastic predator-prey model with disease in prey population

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented.

DETERMINISTIC PREDATOR–PREY MODELS WITH DISEASE IN THE PREY POPULATION

A class of deterministic predator–prey systems, where the prey population is subject to an infectious disease, is studied. The disease can be transmitted both horizontally and vertically within the host population but cannot be spread between the two trophic levels. Using the mathematical tools of uniform persistence, we derive sufficient conditions for which the interacting populations can coexist. Criteria based on model parameters for which either only the infected prey or healthy prey persists are also provided. It is found through numerical investigations that predation can change competition outcomes between healthy and infected prey populations. Depending on the parameter regimes and initial conditions, predation can either eradicate or promote the disease. As infected prey provides a resource for the predators, disease may promote persistence of the predators. However, infected prey may dominate the predator–prey community if disease is very infectious. In addition, disease in the prey population can mediate a hydra effect in predators and may induce chaotic interactions.

STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

Some fractal thoughts about the COVID-19 infection outbreak

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Criticism on the “Mass Hypothesis” in models of epidemic spreading.

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Review of the Generalized Richards Model of epidemic outbreaks.

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Review of the form of segregation named “herd behaviour” in the Trophic Web Theory.

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Adaptation of “herd behaviour” to epidemic outbreak description, as a motivation to the Generalized Richards Model.

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Interpretation of the exponents p and α of the Generalized Richards Model with particular reference to the COVID-19 case.

Some ideas are presented about a geometric motivation of the apparent capacity of generalized logistic equations to describe the outbreak of quite many epidemics, possibly including that of the COVID-19 infection. This interpretation pivots on the complex, possibly fractal, structure of the locus describing the “contagion event set”, and on what can be learnt from the models of trophic webs with “herd behaviour”.

Under the hypothesis that the total number of cases, as a function of time, is fitted by a solution of the Generalized Richards Model, it is argued that the exponents appearing in that differential equation, usually determined empirically, represent the geometric signature of the non-space filling, network-like locus on which contagious contacts take place.

Consumer-resource coexistence as a means of reducing infectious disease

Maintaining sustainable ecosystems are important for all the inhabitants of earth. Also, an important component of sustainable ecosystems is the maintenance of healthy coexistence of consumers and their resources which can include diseases in the species involved. We formulate a model, where the resources are plants, to explore how consumer-resource coexistence could of itself limit the spread of infectious diseases. The important mathematical features of the model are discussed using the basic reproduction number and the consumption number. The results show an association between species coexistence and a decrease in ecosystem resource disease. The possible importance of these results are discussed.

Predator-prey equations with constant harvesting and planting

We propose Lotka-Volterra type predator-prey equations which include small constant terms. Depending on its sign, the constant may model various things. To see the effect of the constants clearly we drop all other functional responses except the ones in the original Lotka-Volterra equations. We add a small negative constant for the harvesting or the Allee effect. A positive constant is added to model the planting or external influx. We find the predator-prey equations with constant terms produce most of dynamic and static patterns observed from other predator-prey models with various functional responses.

SOCIAL BEHAVIOR OF GROUP DEFENSE IN A PREDATOR–PREY SYSTEM WITH DELAY

This paper proposes a mathematical model describing the dynamics of both Antarctic krill and Blue whale population where Antarctic krill species (prey) has group defense when they aggregate in herds in order to provide a self-defense from Blue whale species (predator). Firstly, a sufficient condition for the global asymptotical stability of the positive equilibrium is given using Poincaré–Bendixson theorem excluding periodic solution. Furthermore, we discuss the existence of Hopf-bifurcation and give criteria for stability switches through the study of a first-order exponential polynomial characteristic equation. Finally, some numerical simulations are given to illustrate our results.