Food-limited Model Research Articles

On the Geometric Diversity of Wavefronts for the Scalar Kolmogorov Ecological Equation

In this work, we answer three fundamental questions concerning monostable travelling fronts for the scalar Kolmogorov ecological equation with diffusion and spatiotemporal interaction: these are the questions about their existence, uniqueness and geometric shape. In the particular case of the food-limited model, we give a rigorous proof of the existence of a peculiar, yet substantive non-linearly determined class of non-monotone and non-oscillating wavefronts. As regards to the scalar models coming from applications, this kind of wave solutions is analyzed here for the first time.

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

AbstractWe are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of ‘sufficiently small parameters’, our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al., we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.

Hopf bifurcation in a delayed food-limited model with feedback control

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.

Stability and Neimark—Sacker bifurcation analysis of a food-limited population model with a time delay

In this paper, a kind of discrete delay food-limited model obtained by the Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, the linear stability of this model is studied. It is shown that Neimark—Sacker bifurcation occurs when τ crosses certain critical values. The explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form. Finally, numerical simulations are performed to verify the analytical results.

Existence and Asymptotic Behavior of Positive Solutions of a Non-Autonomous Food-Limited Model with Unbounded Delay

Consider the non-autonomous logistic model \Delta x_n = p_n x_n \frac{1–x_n–k_n}{1+\lambda x_n–k_n}^r \;\;\;(n≥0) where \Delta x_n = x-{n+1} – x_n , { p_n } is a sequence of positive real numbers, { k_n } is a sequence of non-negative integers such that { n–k_n } is non-decreasing, \lambda \in [0,1] , and r is the ratio of two odd integers. We obtain new sufficient conditions for the attractivity of the equilibrium x = 1 of the model and conditions that guarantee the solution to be positive, which improve and generalize some recent results established by Phios and by Zhou and Zhang.

Periodic solutions of periodic generalized food limited model

By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a periodic generalized food limited model with state dependent delays and distributed delays is studied, respectively.