Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion

This paper is concerned with the diffusive Leslie–Gower prey–predator model with weak diffusion. Assuming that the diffusion rates of prey and predator are sufficiently small and the natural growth rate of prey is much greater than that of predators, the diffusive Leslie–Gower prey–predator model is a singularly perturbed problem. Using travelling wave transformation, we firstly transform our problem into a multiscale slow-fast system with two small parameters. We prove the existence of heteroclinic orbit, canard explosion phenomenon and relaxation oscillation cycle for the slow-fast system by applying the geometric singular perturbation theory. Thus, we get the existence of travelling waves and periodic solutions of the original reaction–diffusion model. Furthermore, we also give some numerical examples to illustrate our theoretical results

Dynamics for the diffusive Leslie-Gower model with double free boundaries

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to\infty$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u,v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.Comment: 19 page

Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay

This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings.&nbsp

KESTABILAN MODEL MANGSA PEMANGSA DENGAN BENTUK LESLIE-GOWER STRUKTUR PADA MANGSA

Along with the development of technology so rapidly the development of knowledge about dynamic systems is also growing rapidly.This aims tostudy find out the stability of prey-predator model using Leslie-Gower from with stages of structure on prey. Issues raised in the research are how to generate a mathematical model of the modification of Leslie-Gower prey-predator system with the response function of Holling type II; how to determine the point of equilibrium and stability analysis on a modified model of Leslie-Gower prey-predator with the response function of Holling type II; the effects of changes in the parameters on the actual state of the modification of Leslie-Gower prey-predator model with the response function og Holling type II; and the numerical simulation of the modification of Leslie-Gower prey-predator model with the response function oh Holling type II using the maple software. The problem was analysed based on a literature review. The steps used were: generating a mathematical model from the modification of Leslie-Gower prey-predator model with the response function of Holling type II; determine all fixed point; determine a cracteristic equation and the eigen value of jacobian matriks; determine in the value of the parameter of Hoft bifurcation; calculate the transversal angles; createa numerical simulation of the modification of Leslie-Gower prey-predator model with the response function of Holling type II using the software Maple; and drawing the conclusion.The results of the research show a model as follows:Seiring dengan perkembangan teknologi yang begitu pesat maka perkembangan pengetahuan tentang sistem dinamik juga berkembang pesat. Penelitian ini bertujuan menemukan kestabilan model mangsa pemangsa dengan bentuk Leslie-Gower dengan tahapan struktur pada mangsa. Permasalahan yang diangkat dalam penelitian ini adalah bagaimana menurunkan model matematika dari modifikasi sistem mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II; bagaimana menentukan titik kesetimbangan dan analisis kestabilan pada modifikasi model mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II; bagaimana pengaruh parameter terhadap keadaan yang sebenarnya dari modifikasi model mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II; dan bagaimana simulasi numerik dari modifikasi model mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II menggunakansoftware Maple. Metode yang digunakan untuk menganalisis masalah adalah dengan studi pustaka. Langkah-langkah yang digunakan adalah menurunkan model matematika dari modifikasi sistem mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II, menentukan semua titik tetap, menentukan persamaan karakteristik dan nilai eigen dari matriks jacobian, menentukan nilai parameter terjadinya bifurkasi Holf, menghitung syarat transversal, membuat simulasi numerik dari modifikasi model mangsa pemangsa Leslie-Gower dengan fungsi respon Holling Tipe II menggunakan software Maple dan menyimpulkan.Hasil penelitian memperlihatkan model yang diperoleh adala