انشعابات هم بعد - 2 سیستم گسسته‌ شکار و شکار‌چی

در این مقاله به بررسی رفتار‌های دینامیکی یک سیستم گسسته‌ شکار و شکارچی می‌پردازیم. وجود و پایداری نقاط ثابت سیستم را بررسی می‌کنیم و شرایط کافی برای وجود انشعاب فیلیپ و نایمارکساکر را ارائه می‌دهیم. با استفاده از روش‌های عددی انشعاب و جعبه ابزار  matcontmخم های انشعاب نقطه ثابت از قبیل خم‌ انشعاب نایمارک - ساکر را به همراه نقاط انشعاب روی این خم‌ها به‌دست آورده و  سیکل‌های تا تکرار 32 را محاسبه می‌کنیم. تمام انشعابات هم بعد -1 و هم بعد - 2 و سوئیچ  انشعاب های ‌هم بعد 2 را هم محاسبه می کنیم و در نهایت با استفاده از شبیه سازی عددی رفتار آشوبی سیستم را نمایش می دهیم. 

codimension-2 bifurcations of a discrete-time predator-prey system

introduction  in population dynamics, discretetime dynamical systems have been used to describe interaction between ecological species. comparing to continuoustime dynamical systems, discretetime models are more suitable to describe populations with non overlapping generations. these models in general produce rich and complex dynamical behaviors.among various population interaction, predatorprey models play a fundamental rule in mathematical ecology. the dynamics of predatorprey system is greatly depend on the implementation of the functional response, the availability of prey for predation. in this paper we consider a planar system which describes a predatorprey model. in order to reveal comprehensive dynamics of the system, we employee theoretical tools such as center manifold theorem along with numerical tools based on numerical continuation method. material and methodsour analysis is based on theoretical and numerical techniques. we first determine all fixed points of the system and conditions under which these points may undergo different bifurcations. to reveal more dynamics of the system, we also use numerical bifurcation methods and numerical simulations, which further examine the obtained analytical results.results and discussionfor the resented discretetime predatorprey system, we compute several bifurcation curves, all possible codimension1 and codimension2 bifurcations on thses curves along with their corresponding normal form coefficients. by branch switching technique and employing software package matcontm, we compute stability boundaries for several cycles up to period 32. we also use numerical simulation, to compute basin of attraction and strange attractor emerging around a neimarksacker bifurcation. conclusionwe can highlight the following results from this paper.detection and location of all fixed points of a discretetime predator pray system.computing all possible codimension1 and 2 bifurcation and their corresponding normal form coefficients which in turn reveal criticality of the bifurcation points and determine if extra bifurcation curves can emanate from each detected bifurcation.computing orbits up to period 32 which determine stability thresholds for different cycles.computing basin of attraction and strange of attractor which emerge around a neimarksacker bifurcation by means of numerical simulation technique.