Spatial Dynamics of Some Evolution Systems in Biology

Abstract

We first give a brief review on traveling waves, spreading speeds, and global stability for monotone evolution systems with monostable and bistable nonlinearities. Then we outline our recently developed theory and methods for general monotone semi flows and certain non-monotone systems, and their applications to some biological models.

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

“…In both cases, (S) is uniformly globally well-posed on U 0+ . However, for p, q ≥ 1, u(x, t) → 1 as t → ∞ through the propagation of finite speed travelling wave structures [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], whereas, for 0 < p, q < 1, u(x, t) → 1 uniformly for x ∈ R (through uniform terms of O(t −1/2 ) as t → ∞), as demonstrated in this paper. In fact, we can now immediately infer stability…”

“…In particular, classical Hadamard well-posedness has been established, along with considerable qualitative information regarding the structure of the solution to (1.1)-(1.3). Specific attention has been focused on the convergence to the equilibrium state u = 1 via the evolution of travelling wave structures in the solution to (1.1)-(1.5) when the initial data is non-trivial, as t → ∞, their propagation speed, shape and form [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The cases when 0 < p < 1 and/or 0 < q < 1 have received much less attention, primarily because the nonlinearity f : R → R lacks Lipschitz continuity in these cases owing to the behaviour at u = 0 and/or u = 1 and the classical comparison theorems and continuous dependence theorems fail to apply.…”

Minimal wave speed in a dispersal predator–prey system with delays

“…In the past decades, some important results were established for monotone semiflows; see [1][2][3][4][5][6] and a survey paper by Zhao [7]. In particular, there are some important thresholds that have been widely and intensively studied, and one is the minimal wave speed of traveling wave solutions, which plays an important role modeling biological processes and chemical kinetic [8,9].…”

Spreading speeds of a Lotka–Volterra predator–prey system: The role of the predator

“…On the spreading speed of evolutionary systems, there are many important results; see [17,19] and a survey paper by Zhao [20]. From the viewpoint of monotone dynamical systems, these results were established for the cooperative systems.…”