Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay

Abstract

In this paper, a prey-predator model and weak Allee effect in prey growth and its dynamical behaviors are studied in detail. The existence, boundedness, and stability of the equilibria of the model are qualitatively discussed. Bifurcation analysis is also taken into account. After incorporating the searching delay and digestion delay, we establish a delayed predator-prey system with Allee effect. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. Finally, we present some numerical simulations to illustrate our theoretical analysis.

Quasilinear elliptic equations with exponential nonlinearity and measure data

“…Let and complementary function to Q p ( s ) defined as . An elementary property of functional Q p ( s ) and is the Young inequality: .…”

“…For more related results, one can refer to the previous studies of these and other related results. [11][12][13][14][15][16][17][18][19][20][21][22][23] Before discussing related results of Equation (1) with exponential nonlinearity in details, we would like to recall some definitions. For > 0, p > 1 such that p < N, the R-truncated Wolff potential W R ,p (x) of a nonnegative Borel measure on R N is defined by…”

“…and complementary function to Q p (s) defined as Q * p (r) = max{rs − Q p (s) ∶ s ≥ 0}. An elementary property of functional Q p (s) and Q * p (r) is the Young inequality 23 : sr ≤ Q p (s) + Q * p (r), s, r > 0. Define the corresponding Bessel and Riesz capacities respectively by…”

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…For other closely related results of Equation (1.2) with g ≡ 0, please see the classical works. [3][4][5][6][7][8][9][10][11] There are some results about the Harnack-type inequality present in the literature for nonlocal fractional elliptic problem (1.1). In Caffarelli et al, 12 via the extension formulations for fractional Laplacian, 13 moving spheres method, and blow-up analysis, they established the Harnack-type inequality of positive solutions to { −div(t 1−2 ∇U) = 0,…”

Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response

“…In contrast, populations with weak Allee effects do not have this threshold. Recently, researchers have focused on the Allee effect on different ecosystems, including discrete-time systems [2][3][4][5][6][7] and continuous-time systems [8][9][10][11][12][13][14]. Zhao and Lv [15] study the dynamic complexity of a host-parasitoid system with a lower bound for host, and the form of Allee effect is H(t)-n H(t)+m , where m is an Allee effect constant, n is the lower bound for the host.…”