Traveling wave solutions of a nonlocal dispersal predator–prey model with spatiotemporal delay

Existence of Traveling Waves for Ratio-dependent Predator-prey System with Nonlocal Diffusion

“…If Å #, 0 is uniformly continuous and bounded, and Å #, satisfies Proof. By contradiction, we suppose that there exists some U 8 < U * such that system (5) has a positive solution V , V satisfying (31). Then V is bounded on ℝ.…”

“…Some previous work for dynamics of diffusive Holling-Tanner predator-prey systems on a bounded region can be found in [3,25]. Zuo and Shi in [31] has studied the reactiondiffusion Holling-Tanner type predator-prey system with ratio-dependent functional response ! " #, = $ %% #, + #, 1 − #, − #, #, #, + ' #, , #, = %% #, + ( #, 1 − #, ) #, , the two species move randomly along a one-dimensional region R, and the parameter $ > 0 is a rescaled diffusion coefficient of the prey species while the diffusion coefficient for the predator is rescaled to be 1.…”

“…If the preys are only as food for the predator, that is, ≡ 1 , then the second equation of system (4) is simplified to fisher's KPP equation. Some authors obtained the nonexistence of traveling wave solutions of predator systems by considering the related Cauchy problem of fisher's equation, see [31]. Throughout this paper, we need the below assumptions of the kernel function <.…”

Traveling wavefronts for density‐dependent diffusion reaction convection equation with time delay

“…Considerable evidence shows that traveling wave solution may determine the long-term behaviors of other solutions, so it plays a crucial role in biological invasion and epidemic spreading. Quite a few powerful techniques have been proposed to tackle the existence of traveling waves for density-dependent diffusion reaction equations with small time delay, such as the phase-plane technique and the maximum principle for parabolic equations; see previous works [1][2][3][4][5][6][7] and the references therein.…”

Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

“…The incidence rate in these two papers is bilinear form βSI. Since then, there has been extensive research with traveling wave solutions for delayed diffusive SIR models with various incidence rates, such as βSI 1+αI , βI p S q (p, q > 0) and general form f (S)g(I), see [3][4][5][6][7][8][9][10][11][12][13][14]. There have also been some papers about traveling wave solutions for delayed diffusive SIR models with external supplies [15,16] and delayed diffusive SIRS epidemic models [17,18].…”