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

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

The basins of attraction in a modified May-Holling-Tanner predator–prey model with Allee affect

I analyse a modified May–Holling–Tanner predator–prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator–prey population. 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 Bogdanov–Takens bifurcations. We use simulations to illustrate the behaviour of the model.</p

Temporal and spatio-temporal dynamics in predator-prey models

This thesis analyses temporal and spatio-temporal modified Holling-Tanner predator-prey models with alternative food for predators. Different types of functional responses and a density-dependent phenomenon called Allee effect(s) on the prey were considered. By using analytical and numerical analysis, the stability of the equilibrium points for the different combinations of modifications were proven. The necessary conditions for the models to undergo a different type of bifurcation were illustrated. Additionally, this thesis has provided numerical evidence where the Turing instability leads to spatio-temporal patterns in a specific version of the model

Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator

A predator–prey model with functional response Holling type II, Allee effect in the prey and a generalist predator is considered. It is shown that the model with strong Allee effect has at most two positive equilibrium points in the first quadrant, one is always a saddle point and the other exhibits multi-stability phenomenon since the equilibrium point can be stable or unstable. The model with weak Allee effect has at most three positive equilibrium points in the first quadrant, one is always a saddle point and the other two can be stable or unstable node. In addition, when the parameters vary in a small neighbourhood of system parameters the model undergoes different bifurcations, such as saddle–node, Hopf and Bogdanov–Takens bifurcations. Moreover, numerical simulation is used to illustrate the impact in the stability of positive equilibrium point(s) by adding an Allee effect and an alternative food sources for predators.</p

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

In this manuscript, we study a Leslie–Gower predator-prey model with a hyperbolic functional response and weak Allee effect. The results reveal that the model supports coexistence and oscillation of both predator and prey populations. We also identify regions in the parameter space in which different kinds of bifurcations, such as saddle-node bifurcations, Hopf bifurcations and Bogdanov–Takens bifurcations.</p

Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response

We study the Bazykin predator-prey model with predator intraspecific interactions and ratio-dependent functional response and show the existence and stability of two interior equilibrium points. We prove that the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and Bogdanov-Takens bifurcations. We use numerical simulations to further illustrate the impact changing the predator per capita consumption rate has on the basin of attraction of the stable equilibrium points, as well as the impact of changing the efficiency with which predators convert consumed prey into new predators.</p