Predator‐prey models with memory and kicks: Exact solution and discrete maps with memory

Abstract

In this paper, we proposed new predator‐prey models that take into account memory and kicks. Memory is understood as the dependence of current behavior on the history of past behavior. The equations of these proposed models are generalizations of the Lotka‐Volterra and Kolmogorov equations by using the Caputo fractional derivative of non‐integer order and periodic kicks. This fractional derivative allows us to take into account memory with power‐law fading. The periodic kicks, which are described by Dirac delta‐functions, take into account short duration of interaction between predators and prey. For the proposed equations, which are fractional differential equations with kicks, we obtain exact solutions that describe behaviors of predator and prey with power‐law fading memory. Using these exact solutions, we derive, without using any approximations, new discrete maps with memory that represent the proposed predator‐prey models with memory.

General Fractional Dynamics

“…Note that the proposed equations and mappings can be used to describe economic processes with memory [17,47], for non-Markovian quantum processes [48], processes in the dynamics of populations [49], and many other processes.…”

“…(3) Another way of describing general fractional dynamics with discrete time can be based on the use of discrete mappings obtained from exact solutions of general fractional differential and integral equations with periodic kicks. For the first time discrete mappings with nonlocality in time were derived from fractional differential equations in [43][44][45], in which the Riemann-Liouville and Caputo fractional derivatives were used (see also [2,[46][47][48][49]). The proposed approach allows us to derive discrete time mappings with nonlocality in time from integro-differential equations of non-integer orders without approximation.…”

“…, X 1 ), where the number of variables in the function F increases with the number n ∈ N. For the first time such discrete mappings were derived from fractional differential equations in [43]. Then, this approach was developed in [2,44,45], and it was applied in [17,[46][47][48][49][50]. We should emphasize that nonlocal mappings were derived from fractional differential equations without approximations.…”

Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

“…The proposed nonlocal maps can be derived by using the equivalence of the FDE and the Volterra integral equations in [193,194,109]. Then, this approach has been applied in [195,196,182,197,107,103,198,199]. The first computer simulations of the suggested nonlocal maps were made in [200,201], and then numerical simulations have proved the existence of new types of attractors and new types of chaotic behavior for these maps (see [202,203,204,205,206] and references therein).…”

“…We should note that the discrete maps with memory and nonlocality in time, which are derived from fractional differential equations [192,193,194,109,195,196,182,197,199,107], can demonstrate qualitatively new types of chaotic and regular behavior. Similarly, fundamentally new types of self-organization are expected in processes with nonlocality in time and memory.…”