Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented

Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented

Impact of alternative food on predator diet in a Leslie-Gower model with prey refuge and Holling Ⅱ functional response

Since certain prey hide from predators to protect themselves within their habitats, predators are forced to change their diet due to a lack of prey for consumption, or on the contrary, subsist only with alternative food provided by the environment. Therefore, in this paper, we propose and mathematically contrast a predator-prey, where alternative food for predators is either considered or not when the prey population size is above the refuge threshold size. Since the model with no alternative food for predators has a Hopf bifurcation and a transcritical bifurcation, in addition to a stable limit cycle surrounding the unique interior equilibrium, such bifurcation cases are transferred to the model when considering alternative food for predators when the prey size is above the refuge. However, such a model has two saddle-node bifurcations and a homoclinic bifurcation, characterized by a homoclinic curve surrounding one of the three interior equilibrium points of the model

Una clase de modelo de depredación del tipo Leslie-Gower con respuesta funcional racional no monotónica y alimento alternativo para los depredadores

The interactions between two species are basic in the study of complex food chains, particularly the relation among the predators and their prey.The analysis of simple models, described by continuous-time systems, in which some ecological phenomena are incorporated giving lights about this interesting interrelationship.In this work, a Leslie-Gower type predator-prey model is analyzed, considering two aspects: the prey defends from the predation, forming group defense and the predators have an alternative food. So, a rational Holling type IV functional response and a modification of the predators carrying capacity are assumed, to describe each phenomenon.We determine conditions on the parameter space for the existence of the equilibria and their nature.Using the Lyapunov quantities method, we also establish conditions on the parameter values for which there exist a unique positive equilibrium point, which is stable and surrounded by two limit cycle, the innermost unstable and the outermost sable.We conclude that the parameter indicating the existence of alternative food for predator has a great importance on the dynamic of model, because appear new equilibrum points and separatrix curves in the phase plane.Some simulations are given to reinforce our findings the ecological interpretations of resultas are given.Las interacciones entre dos especies son básicas en el estudio de cadenas alimentarias complejas, en particular la relación entre los depredadores y sus presas.El análisis de modelos simples, descritos por sistemas de tiempo continuo, en los cuales se incorporan algunos fenómenos ecológicos dando luces sobre esta interesante interrelación.En este trabajo, se analiza un modelo de depredador-presa del tipo Leslie-Gower, descrito por un sistema de ecuaciones diferenciales ordinarias (EDO) considerando dos aspectos: la presa se defiende de la depredación, formando grupo de defensa, y los depredadores disponen un alimento alternativo, cuando su alimento favorito escasea. Por lo tanto, se asume una respuesta funcional racional de Holling tipo IV y una modificación de la capacidad de carga de los depredadores para describir estos fenómenos.Determinamos las condiciones en el espacio de parámetros para la existencia de los equilibrios y la naturaleza de cada uno de ellos.Concluimos que el parámetro que indica la existencia de alimento alternativo para depredadores tiene una gran importancia en la dinámica del modelo, porque aparecen nuevos puntos de equilibrio y curvas de separatriz en el plano de fase.Por simulaciones numéricas comprobamos que existe un subconjunto de parámetros para los cuales hay un único punto de equilibrio positivo en el plano de fase, el cual es estable y está rodeado por dos ciclos límites originados por bifurcación de Hopf, el interior inestable y el exterior estable

Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response

A thorough analysis is performed in a predator-prey reaction-diffusion model which includes three relevant complex dynamical ingredients: (a) a strong Allee effect; (b) ratio-dependent functional responses; and (c) transport attributes given by a diffusion process. As is well-known in the specialized literature, these aspects capture adverse survival conditions for the prey, predation search features and non-homogeneous spatial dynamical distribution of both populations. We look for traveling-wave solutions and provide rigorous results coming from a standard local analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections and periodic orbits in the associated dynamical system in $R^4$. In so doing, we present and describe a diverse zoo of traveling wave solutions; and we relate their occurrence to the Allee effect, the spreading rates and propagation speed. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to enravel a multiplicity of wave solutions in the model. A deep understanding of such ecological dynamics is therefore highlighted.Comment: 35 pages, 22 figure

Pattern Formation in a Bacterial Colony Model

We investigate the spatiotemporal dynamics of a bacterial colony model. Based on the stability analysis, we derive the conditions for Hopf and Turing bifurcations. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by parameters in the model and find that the model dynamics exhibit a diffusion controlled formation growth to spots, holes and stripes pattern replication, which show that the bacterial colony model is useful in revealing the spatial predation dynamics in the real world

Dynamic analysis of a Leslie-Gower predator-prey model with the fear effect and nonlinear harvesting

In this paper, we investigate the stability and bifurcation of a Leslie-Gower predator-prey model with a fear effect and nonlinear harvesting. We discuss the existence and stability of equilibria, and show that the unique equilibrium is a cusp of codimension three. Moreover, we show that saddle-node bifurcation and Bogdanov-Takens bifurcation can occur. Also, the system undergoes a degenerate Hopf bifurcation and has two limit cycles (i.e., the inner one is stable and the outer is unstable), which implies the bistable phenomenon. We conclude that the large amount of fear and prey harvesting are detrimental to the survival of the prey and predator