A Holling-Tanner predator-prey model with strong Allee effect

We analyse a modified Holling-Tanner predator-prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements results of previous articles by Saez and Gonzalez-Olivares (SIAM J. Appl. Math. 59 1867-1878, 1999) and Arancibia-Ibarra and Gonzalez-Olivares (Proc. CMMSE 2015 130-141, 2015)discussing Holling-Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to co-existence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogadonov-Takens bifurcations

Estabilidad de un modelo depredador-presa tipo Leslie Gower con un efecto Allee fuerte con retardo

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.En este trabajo se estudia un modelo depredador-presa del tipo Leslie-Gower modificado que introduce en el crecimiento de la población de presas un fuerte efecto Allee retardado.El modelo Leslie-Gower con efecto Allee no tiene ninguno, uno o dos puntos de equilibrio positivos, pero la incorporación de un retardo temporal en la tasa de crecimiento desestabiliza el sistema, rompiendo la estabilidad cuando el retardo cruza un punto crítico. Se estudia en detalle la existencia de una bifurcación de Hopf y las simulaciones numéricas confirman los resultados teóricos mostrando los diferentes escenarios. Presentamos interpretaciones biológicas para especies de tipo presa-predado

Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (with- out) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings

Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings

The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model

In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results

Modelling and analysis of a modified May-Holling-Tanner predator-prey model with Allee effect in the prey and an alternative food source for the predator

In the present study, we have modified the traditional May-Holling-Tanner predator-prey model used to represent the interaction between least weasel and field-vole population by adding an Allee effect (strong and weak) on the field-vole population and alternative food source for the weasel population. It is shown that the dynamic is different from the original May-Holling-Tanner predator-prey interaction since new equilibrium points have appeared in the first quadrant. Moreover, the modified model allows the extinction of both species when the Allee effect (strong and weak) on the prey is included, while the inclusion of the alternative food source for the predator shows that the system can support the coexistence of the populations, extinction of the prey and coexistence and oscillation of the populations at the same time. Furthermore, we use numerical simulations to illustrate the impact that changing the predation rate and the predator intrinsic growth rate have on the basin of attraction of the stable equilibrium point or stable limit cycle in the first quadrant. These simulations show the stabilisation of predator and prey populations and/or the oscillation of these two species over time.Comment: 18 pages, 8 figure

Effect of Fear in Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response Incorporating Prey Refuge

In the present paper, we study the effect of antipredator behavior due to fear of predation on a modified Leslie-Gower predator-prey model incorporating prey refuge which predation rate of predators follows Beddington-DeAngelis functional response. The biological justification of the model is demonstrated through non-negativity, boundedness, and permanence. Next, we perform the analysis of equilibrium and local stability. We obtain four equilibrium points where two points are locally asymptotically stable and other points are unstable. Besides, we show the effect of the fear in the model and obtain a conclusion that the increased rate of fear can decrease the density of both populations, and prey populations become extinct. Meanwhile, for the case with a constant rate of fear, the prey refuge helpful to the existence of both populations. However, for the case with the fear effect is large, prey refuge cannot cause the extinction of predators. Several numerical simulations are performed to support our analytical results