Delayed Population Model Research Articles

Technique to derive discrete population models with delayed growth

We provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles that a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of, what we consider, the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to obtain two models, a Beverton–Holt adult model and a Ricker adult model and discuss the global dynamics of both models.

Monotone reducing mechanism in delayed population model with degenerate diffusion

In this paper, we study the monotone reducing mechanism of degenerate diffusion equations with time delay. We show the monotone dependence of critical wave speed on time delay. Due to the complex dynamics arising from degeneracy and time delay, we use a new phase transform approach to analyze the delicate local and global behaviors of critical sharp traveling waves and then derive the comparison of critical speeds.

Analysis and control of a delayed population model with an Allee effect

Based on the population model with an Allee effect, the corresponding delayed model is built up. According to the qualitative theory of differential equations, the model is analyzed completely to describe various intriguing dynamic properties as singular point of high order, Hopf bifurcation, etc. Next, combined impulse control theory with the ecological principles, the control models with impulse effects are proposed and the corresponding controllers are designed not only to avoid the extinction of the populations but also to prevent outbreak of the populations. The study would not only reveal the complexity dynamic behavior of ecosystems, but also provide the theory reference for the management of populations.

Accuracy of environmental tracers and consequences for determining the Type Ia supernova magnitude step

Type Ia Supernovae (SNe Ia) are standardizable candles that allow us to measure the recent expansion rate of the Universe. Due to uncertainties in progenitor physics, potential astrophysical dependencies may bias cosmological measurements if not properly accounted for. The dependency of the intrinsic luminosity of SNe Ia with their host-galaxy environment is often used to standardize SNe Ia luminosity and is commonly parameterized as a step function. This functional form implicitly assumes two-populations of SNe Ia. In the literature, multiple environmental indicators have been considered, finding different, sometimes incompatible, step function amplitudes. We compare these indicators in the context of a two-populations model, based on their ability to distinguish the two populations. We show that local Hα-based specific star formation rate (lsSFR) and global stellar mass are better tracers than, for instance, host galaxy morphology. We show that tracer accuracy can explain the discrepancy between the observed SNe Ia step amplitudes found in the literature. Using lsSFR or global mass to identify the two populations can explain all other observations, though lsSFR is favoured. As lsSFR is strongly connected to age, our results favour a prompt and delayed population model. In any case, there exists two populations that differ in standardized magnitude by at least 0.121 ± 0.010 mag.

Hopf Bifurcation in a Delayed Population Model Over Patches with General Dispersion Matrix and Nonlocal Interactions

Hopf Bifurcation in a Delayed Population Model Over Patches with General Dispersion Matrix and Nonlocal Interactions

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.

Global attractors, extremal stability and periodicity for a delayed population model with survival rate on time scales.

In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of the delayed population model with survival rate on isolated time scales given by $ x^{\Delta} (t) = \gamma(t) x(t) + \dfrac{x(d(t))}{\mu(t)}e^{r(t)\mu(t)\left(1 - \frac{x(d(t))}{\mu(t)}\right)}, \ \ t \in \mathbb T. $ We present many examples to illustrate our results, considering different time scales.

Pseudo compact almost automorphic solutions for a family of delayed population model of Nicholson type

In this paper, we consider a generalized differential equation with feedback and time-varying delays and prove the existence of a unique pseudo compact almost automorphic solution that is exponentially attractive. We show our result by applying the Banach fixed point theorem and the properties of pseudo compact almost automorphic function. We apply our results and obtain new criteria for the existence and convergence dynamics of pseudo compact almost automorphic solutions for some models that involves sum of nonlinear terms that satisfy a general condition. These models are very general and include sums of unimodals terms with monotones, such as Nicholson, Mackey-Glass, Lasota-Wazeska and others. The results obtained are new and they recover, extend and improve recent works.

Stable analysis for a delayed population model with survival rate

Stable analysis for a delayed population model with survival rate

Traveling Waves Connecting Equilibrium and Periodic Orbit for a Delayed Population Model on a Two-Dimensional Spatial Lattice

This paper is concerned with the existence of fast traveling waves connecting an equilibrium and a periodic orbit in a delayed population model with stage structure on a two-dimensional spatial lattice, under the assumption that the corresponding ODEs have heteroclinic orbits connecting an equilibrium point and a periodic solution. In this work, we rewrite the mixed functional differential equation as an integral equation in a Banach space and analyze the corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by using the Liapunov–Schmidt method and implicit function theorem.

On a nonlinear delay population model

The nonlinear delay differential equation x˙(t)=r(t)[g(t,xt)−h(x(t))],t≥0 is considered. Sufficient conditions are established for the uniform permanence of the positive solutions of the equation. In several particular cases, explicit formulas are given for the upper and lower limit of the solutions. In some special cases, we give conditions which imply that all solutions have the same asymptotic behavior, in particular, when they converge to a periodic or constant steady-state.

Delayed population models with Allee effects and exploitation.

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

The influence of time-dependent delay on behavior of stochastic population model with the Allee effect

This paper presents the analysis of behavior of stochastic time-dependent delay population model with the Allee effect. We prove the existence-and-uniqueness of positive solution of considered model. Then, we find the sufficient conditions under which the population will become extinct. We also show that if the initial population size exceeds environmental carrying capacity and time delay is sufficiently long, considered population is non-persistent in mean. The sufficient conditions for asymptotical mean square stability and stability in probability of the positive equilibrium states of the model, in terms of Lyapunov functional method, are obtained. Finally, as an illustration, we apply our mathematical results and predict time which a population of the African wild dog Lycaon pictus needs to reach it is equilibrium states, and also confirm that population of brown tree snake Boiga irregularis is non-persistent in mean if the initial population size is greater than carrying capacity and time delay is long enough.

On a delay population model with a quadratic nonlinearity without positive steady state

A population model described by a nonlinear delay differential equation with a quadratic nonlinearityẋ(t)=∑k=1mαk(t)x(hk(t))-β(t)x2(t),t⩾0is considered where m⩾1 is an integer, functions αk,β:[0,∞)→(0,∞) are continuous, functions hk:[0,∞)→R are continuous such that t-τ⩽hk(t)⩽t,τ=const,τ>0, and, for any t⩾0, the inequality hj(t)<t holds for at least one index j∈{1,…,m}.Although this equation does not have a positive steady state, a new method not based on the existence of a positive steady state is developed and used to investigate the permanence, global attractivity conditions and nonoscillation properties.

Existence and uniqueness of asymptotically constant or periodic solutions in delayed population models

By means of fixed point theory we show that the unique solution of nonlinear difference equations of the formconverges to a predetermined constant or a periodic solution. Then, we show the solution is stable and that its limit function serves as a global attractor. The same theory will be extended to two more models.

Average Conditions for the Permanence of a Bounded Discrete Predator-Prey System

Average conditions are obtained for the permanence of a discrete bounded system with Holling type II functional responseu(n+1)=u(n)exp{a(n)-b(n)u(n)-c(n)v(n)/(u(n)+m(n)v(n))},v(n+1)=v(n)exp{-d(n)+e(n)u(n)/(u(n)+m(n)v(n))}.The method involves the application of estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.

On a delay population model with quadratic nonlinearity

Abstract A nonlinear delay differential equation with quadratic nonlinearity, x ˙ ( t ) = r ( t ) [ ∑ k = 1 m α k x ( h k ( t ) ) − β x 2 ( t ) ] , t ≥ 0 , is considered, where α k and β are positive constants, h k : [ 0 , ∞ ) → R are continuous functions such that t − τ ≤ h k ( t ) ≤ t , τ = const , τ &gt; 0 , for any t &gt; 0 the inequality h k ( t ) &lt; t holds for at least one k, and r : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function satisfying the inequality r ( t ) ≥ r 0 = const for an r 0 &gt; 0 . It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.

Periodic Oscillations in a Delay Population Model

In this paper, a nonlinear delay population model is investigated. Choosing the delay as a bifurcation parameter, we demonstrate that Hopf bifurcation will occur when the delay exceeds a critical value. Global existence of bifurcating periodic solutions is established. Numerical simulations supporting the theoretical findings are included. Keywords—Population model; Stability; Hopf bifurcation; Delay; Global Hopf bifurcation.

Numerical simulations of traveling wave solutions in a drift paradox inspired diffusive delay population model

We describe numerical algorithm for the simulation of traveling wave solutions in a newly formulated drift paradox inspired diffusive delay population model. We use method of lines to discretize the boundary value problem for the reaction-diffusion equations and we integrate in time the resulting system of delay differential equations using the embedded pair of continuous Runge–Kutta methods of order four and three. We advance the solution with the method of order four and the approximations of order three are used for local error estimation. Numerical results demonstrate the robustness, efficiency, and accuracy of our approach. Moreover, these numerical results confirm the recent theoretical results on the minimum traveling wave speed for this model.

Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice

In this paper, we study the existence of traveling waves of a delayed population model with age-structure on a 2-dimensional spatial lattice when the maturation time r is relatively small. Under the assumption that the birth function b satisfies the bistable condition without requiring monotonicity, we prove the persistence of traveling wavefronts by means of a perturbation argument based on the existing results on the asymptotic autonomous system and the Fredholm alternative theory.