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1
artículo
Publicado 2017
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The classic Lotka-Volterra model belongs to a family of differential equations known as “Generalized Lotka-Volterra”, which is part of a classification of four models of quadratic fields with center. These models have been studied to address the Hilbert infinitesimal problem, which consists in determine the number of limit cycles of a perturbed hamiltonian system with center. In this work, we first present an alternative proof of the existence of centers in Lotka-Volterra predator-prey models. This new approach is based in algebraic equations given by Kapteyn, which arose to answer Poincaré’s problem for quadratic fields. In addition, using Hopf Bifurcation theorem, we proof that more realistic models, obtained by a non-linear perturbation of a classic Lotka-Volterra model, also possess limit cycles.
2
artículo
Publicado 2017
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The classic Lotka-Volterra model belongs to a family of differential equations known as “Generalized Lotka-Volterra”, which is part of a classification of four models of quadratic fields with center. These models have been studied to address the Hilbert infinitesimal problem, which consists in determine the number of limit cycles of a perturbed hamiltonian system with center. In this work, we first present an alternative proof of the existence of centers in Lotka-Volterra predator-prey models. This new approach is based in algebraic equations given by Kapteyn, which arose to answer Poincaré’s problem for quadratic fields. In addition, using Hopf Bifurcation theorem, we proof that more realistic models, obtained by a non-linear perturbation of a classic Lotka-Volterra model, also possess limit cycles.
3
artículo
In this work, we establish the way to protect a species (prey) affected by its hunter (predator), using a system of differential equations that models population dynamics. The analysis of this model leads us to guarantee the existence and unique ness of its solutions and through the stability of the system we obtain the conditions for these solutions to be adequate.
4
artículo
In several previous works, different predation models have been modified by considering the use of prey refuges, for which a partial analysis of their dynamics is done. In some of them, it is stated that the use of refuge has a stabilizing effect on the predator-prey interaction. One of the purposes of this paper is to show that some of these new systems, derived from the Lotka-Volterra model, this assertion is not fulfilled. In this work, several of the models studied have more than one positive equilibrium point, and the behavior of the solutions is highly dependent on the initial conditions
5
artículo
This paper deals with a continuous-time predator-prey model of Gause-type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. According to these results, the extinction of both species is not possible, and they coexist over the long term. We conclude that the dynamics of the studied model are very similar to the model that does not consider prey refuge. However, this cannot be stated if another function is used to describe this anti-predator behavior.
6
artículo
Publicado 2022
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This work is intended to provide basic information and numerical experimentation of biological models focusing on how computation can be used to generate results via software R. In addition, this work enriches the scientific literature in Spanish that links mathematics and computational biology. It also provides detailed information on the analysis of Predator-prey, Toxoplasma gondii, and Common influency models. For the development of this article we will speak in the first instance of the dynamics of the predator-prey model. In the last two models, it is solved numerically for a range of values of a given parameter. This in order to show deductions that contribute to a deeper investigation of the data involved, and even to the analysis of a professional specialized in the model.
7
artículo
Publicado 2025
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For many applied mathematicians, and especially for biomathematicians, the first model proposed by the Italian mathematician Vito Volterra in 1926 is well known, describing for the first time the relationship between a predator and its prey. This model coincided with a similar system, on chemical reactions, proposed by the physicist-chemist Alfred J. Lotka years earlier. Since then, and with an epidemic character, variations, modifications, and the incorporation of new phenomena or ecological principles have been formulated to ”make more realistic” the foundations and studies on this fundamental interaction between two species of living beings. In this work, we will give a brief description of the historical context of this seminal model, emphasizing its main properties; then we will add specific modifications, briefly outlining properties of some of them.
8
artículo
This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories. Those trajectories that are above this curve have as their w- limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.
9
artículo
Publicado 2024
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In this work, the well-possednes of a predator-prey model with reaction-diffusion will be studied where the Allee effect in prey reproduction will be included and predation dynamics will be considered a functional response dependent on the species (prey and predator) with the incorporation of the refuge in the dam. In this way, the existence, uniqueness and positivity of the system's solutions as the main result will be guaranteed.
10
artículo
Publicado 2022
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The interactions between predators and their prey in the real world are usually affected by diverse ecological phenomena acting on both the prey and the predators. Collaboration or cooperation between predators is one of those behaviors, which has received less attention from researchers than competition among consumers. These contacts are important aspects of the dynamics of food chains and trophic webs. In this work, we will study the influence of collaboration or cooperation (hunting cooperation) between predators to consume (or capture) their favorite prey, which are affected by an effect Allee weak. We extend the results obtained in a previously published model, considering only collaboration between predators and in which the Allee effect is absent.
11
artículo
Publicado 2021
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The main feature of the Leslie-Gower-type predation model is that the predator’s growth function is one of logistic-type. Thus, it is a model assuming implicitly the competition among predators. In this work the dynamics of a modified Leslie-Gower type predator-prey model is analyzed, considering two important aspects: (i) the predators capture an alternative food when the quantity of prey is scarce and (ii) the prey population is affected by an Allee effect. Considering a topological equivalent system, the main properties of the system are established. Necessary and sufficient conditions for the existence and local stability of equilibria are determined, also showing the existence of a homoclinic orbit and of at least a limit cycle. When the predators are generalists the dynamics of the model differ enough respecting the model considering predators specialist ...
12
artículo
Publicado 2020
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It is well known that predator-prey interactions are strongly dependent on the functional response as well as the population growth rates of prey and predators.In this work, the study will be based on a Leslie-Gower type predation model, described by a twodimensional system of ordinary differential equations (ODEs), assuming the prey population is affected by a strong Allee effect and that predators have an alternative food.The functional response will be assumed linear, which is prey-dependent and monotonously increasing. In turn, the equation of growth of predators will also be considered of the logistic type, where the environmental carrying capacity for predators is assumed proportional to the prey population size. Among the most important results obtained is that for a same set of parameters, there are different behaviors of the system solutions, since two attractor equilibrium poin...
13
artículo
Publicado 2020
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It is well known that predator-prey interactions are strongly dependent on the functional response as well as the population growth rates of prey and predators.In this work, the study will be based on a Leslie-Gower type predation model, described by a twodimensional system of ordinary differential equations (ODEs), assuming the prey population is affected by a strong Allee effect and that predators have an alternative food.The functional response will be assumed linear, which is prey-dependent and monotonously increasing. In turn, the equation of growth of predators will also be considered of the logistic type, where the environmental carrying capacity for predators is assumed proportional to the prey population size. Among the most important results obtained is that for a same set of parameters, there are different behaviors of the system solutions, since two attractor equilibrium poin...
14
artículo
Publicado 2022
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In this paper, a modified Leslie-Gower type predator-prey model introducing in prey population growth a delayed strong Allee effect is studied. Estabilidad de un modelo depredador-presa tipo Leslie Gower con un efecto Allee fuerte con retardo The Leslie-Gower model with Allee effect has none, one or two positive equilibrium points but the incorporation of a time delay in the growth rate destabilizes the system, breaking the stability when the delay cross a critical point. The existence of a Hopf bifurcation is studied in detail and the numerical simulations confirm the theoretical results showing the different scenarios. We present biological interpretations for species prey-predator type.
15
artículo
Publicado 2022
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A predator-prey model of Gause type is an extension of the classical Lotka-Volterra predator-prey model. In this work, we study a predator-prey model of Gause type, where the prey growth rate is subject to an Allee effect and the action of the predator over the prey is given by a square-root functional response, which is non-differentiable at the y-axis. This kind of functional response appropriately models systems in which the prey have a strong herd structure, as the predators mostly interact with the prey on the boundary of the herd. Because of the square root term in the functional response, studying the behavior of the solutions near the origin is more subtle and interesting than other standard models.Our study is divided into two parts: the local classification of the equilibrium points, and the behavior of the solutions in certain invariant set when the model has a strong Allee ef...
16
artículo
Publicado 2022
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In this work, the ecologically well-posedness of a predator-prey model is demonstrated by showing that a region of the first quadrant is a positively invariant attracting set for the solutions of that model. The predator-prey model considers logistic-type growth in both populations and a non-differentiable functional response that generalizes previous ones. Due to non-differentiability, there is no uniqueness of solutions, and the standard methodology cannot be applied directly. Topological equivalences, geometrical arguments, and the Poincare-Bendixson theorem are used to obtain our result.
17
artículo
Publicado 2023
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This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.
18
artículo
Publicado 2022
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In this work, a May-Holling-Tanner ratio-dependent predator-prey model is studied with an alternative food source for the predator, described by a two-dimensional system of ordinary differential equations. We study the existence and uniqueness of the solutions of the mentioned above system. In addition, the boundedness and positivity of these solutions are analyzed and we establish conditions for the local stability of a simplified model, through a differentiable equivalence. Likewise, the Python programming language is used to perform the simulations using the Runge-Kutta numerical method of order four with the aim of showing the different cases of qualitative analysis.
19
artículo
Publicado 2024
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In various previous works, different predation models have been analyzed considering the use of refuge by the prey population, without carrying out an exhaustive analysis of its dynamics. In some of them it is stated that the use of shelters or dens by a fraction of the prey population has a stabilizing effect on predator-prey interaction. One of the objectives of this work is to show that some of these new systems considering refuge may have the same topological portrait in the phase plane as the original; but in others the dynamics change strongly. In our research we will introduce modifications to the well-known Volterra model, considering various ways to express the number of prey in refuge. In several of them, local dynamics equivalent to the original model are obtained, confirming that results reported as new were previously known based on the original system, without takin...
20
artículo
Publicado 2022
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Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs. In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food. We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point. In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.
