Non-autonomous Logistic Equation Research Articles

Positive solutions for the periodic-parabolic problem with large diffusion

<p>In this paper, we study the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. By employing sufficient conditions to guarantee negative principal eigenvalue, we obtain the existence, uniqueness, and stability of the positive periodic solutions. Moreover, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.</p>

On the Random Non-Autonomous Logistic Equation with Time-Dependent Coefficients

In this paper, we deal with the non-autonomous logistic growth model with time-dependent intrinsic growth rate and carrying capacity. Accounting for errors in recorded data, randomness is incorporated into the equation by assuming that the input parameters are random variables. The uncertainty of the model output is quantified by approximations of the first probability density function via the random variable transformation method. A numerical example illustrates the results.

SOLUSI HAMPIRAN PERSAMAAN LOGISTIK NON-AUTONOMOUS

The aim of this study is to derive the approximation solution of the non-autonomous logistic equation with a non-constant carrying capacity. The solution is found via predictor-corrector method (Adams-Bashforth-Moulton method, Milne method and Hamming method). The approximation solution that obtained, then, is compared to the exact solution. The results show that, for small step size, the approximation solution approximate the exact solution is in good agreement.

Stochastic permanence of solution to stochastic non-autonomous logistic equation with jumps

It is investigated the non-autonomous logistic differential equation with disturbance of coeffcients by white noise, centered and non-centered Poisson noises. The coeffcients of equation are locally Lipschitz continuous but do not satisfy the linear growth condition. This equation describes the dynamics of population in the Verhulst model which takes into account the logistic eect: an increase of the population size produces a fertility decrease and a mortality increase; since resources are limited, if the population size exceeds some threshold level, the habitat cannot support the growth. The property of stochastic permanence is desirable since it means the long time survival in a population dynamics. The suffcient conditions for the stochastic permanence of population in the considered model is obtained.

Logistic equation is a simple stochastic carrying capacity

The logistic model has long been used in ecological modelling for its simplicity and effectiveness. Variations on the logistic model are prolific but, to date, there are a limited number of models that incorporate the stochastic nature of the carrying capacity. This study proposes a modification to the logistic model to incorporate a second differential equation which describes the changes in the carrying capacity, thus treating the carrying capacity as a state variable. The carrying capacity is modelled via a stochastic differential equation that accounts for stochastic ('noisy') variations in the finite resources that the population relies on. The extinction probability distribution, expected solution paths, variance of the solution paths, and distribution of the population are computed using the Monte Carlo method. References R. B. Banks. Growth and Diffusion Phenomena . Springer, 1994. doi:10.1007/978-3-662-03052-3 F. Brauer and C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology . Springer, 2001. doi:10.1007/978-1-4614-1686-9 A. Tsoularis and J. Wallace. Analysis of logistic growth models. Math. Biosci. , 179:21–55, 2002. doi:10.1016/S0025-5564(02)00096-2 S. Oppel, G. Hilton, N. Ratcliffe, C. Fenton, J. Daley, G. Gray, J. Vickery and D. Gibbons. Assessing population viability while accounting for demographic and environmental uncertainty. Ecology , 95:1809–1818, 2014. doi:10.1890/13-0733.1 J. E. Cohen. Population growth and earth's human carrying capacity. Science , 269:341–346 1995. doi:10.1126/science.7618100 J. M. Cushing. Periodic time-dependent predator-prey systems. SIAM J. Appl. Math. , 32:82–95 1977. doi:10.1137/0132006 B. D. Coleman. Nonautonomous logistic equation as models of the adjustment of populations to environmental change. Math. Biosci. , 45:159–173, 1979. doi:10.1016/0025-5564(79)90057-9 J. Vandermeer. Seasonal ioschronic forcing of Lotka Volterra equations. Prog. Theor. Phys. , 96:13–28, 1996. doi:10.1143/PTP.96.13 H. M. Safuan, Z. Jovanoski, I. N. Towers and H. S. Sidhu. Exact solution of a non-autonomous logistic population model. Ecol. Model. , 251:99–102, 2013. doi:10.1016/j.ecolmodel.2012.12.016 P. Del Monte-Luna, B. W. Brook, M. J. Zetina-Rejon and V. H. Cruz-Escalona. The carrying capacity of ecosystems. Global Ecol. Biogeogr. , 13:485–495, 2004. doi:10.1111/j.1466-822X.2004.00131.x H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A simple model for the total microbial biomass under occlusion of healthy human skin. 19th International Congress on Modelling and Simulation , Perth, Australia, December 2011, 733–739. http://mssanz.org.au/modsim2011/AA/safuan.pdf H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model. EMAC2011, ANZIAM J. , 53:C172–C184 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. Impacts of biotic resource enrichment on predator-prey population. B. Math. Biol. , 75:1798–1812, 2013. doi:10.1007/s11538-013-9869-7 H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. A two-species predator-prey model in an environment enriched by a biotic resource. CTAC2012, ANZIAM J. , 54:C768–C787, 2014 . http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6376 S. M. Henson and J. M. Cushing. The effect of periodic habitat fluctuations on a nonlinear insect population model. J. Math. Biol. , 36:201–226, 1997. doi:10.1007/s002850050098 V. Mendez, I. Llopis, D. Campos and W. Horsthemke. Extinction conditions for isolated populations affected by environmental stochasticity. Theor. Popul. Biol. , 77:250–256, 2010. doi:10.1016/j.tpb.2010.02.006 P. Foley. Predicting extinction times from environmental stochasticity and carrying capacity. Conserv. Biol. , 8:124–137, 1994. http://www.jstor.org/stable/2386727 D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. , 43:525–546, 2001. doi:10.1137/S0036144500378302 C. W. Gardiner. Handbook of Stochastic Methods , 2nd ed. Springer, 1985. http://trove.nla.gov.au/version/46588286

Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients

In this work we study the asymptotic behaviour of the following prey-predator system \begin{equation*} \left\{ \begin{split} &A'=\alpha f(t)A-\beta g(t)A^2-\gamma AP\\ &P'=\delta h(t)P-\lambda m(t)P^2+\mu AP, \end{split} \right. \end{equation*} where functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are not necessarily bounded above. We also prove the existence of the pullback attractor and the permanence of solutions for any positive initial data and initial time, making a previous study of a logistic equation with unbounded terms, where one of them can be negative for a bounded interval of time. The analysis of a non-autonomous logistic equation with unbounded coefficients is also needed to ensure the permanence of the model.

Exact solution of a non-autonomous logistic population model

Growth models such as the logistic equation are widely studied and applied in population and ecological modelling. The carrying capacity in the logistic equation is usually regarded as a constant which is not often realistic. Functional forms of the carrying capacities are used to describe changes in the environment. The purpose of this study is to derive an exact solution of the non-autonomous logistic equation with a saturating carrying capacity. The solution is found via a power series resulting from a straightforward algebraic method. For practical applications the power series may be truncated, a simple criterion is established that leads to a good approximate solution. The approximate solution is in good agreement with the numerical simulations, even though only a small number of terms are used.

Non-Autonomous Logistic Equations and Optimization of Renewable Resources Management

This paper concerns applications of the non-autonomous logistic equation to bioeconomic fishery models. We develop a generalized model which satisfies both biological and economic principles of optimization without overexploitation. We also provide conditions to optimize the revenue and to maximize the sustainable yield. 16 Copyright © SIAM Unauthorized reproduction of this article is prohibited

Pullback exponential attractors for nonautonomous equations Part II: Applications to reaction–diffusion systems

The existence of a pullback exponential attractor being a family of compact and positively invariant sets with a uniform bound on their fractal dimension which at a uniform exponential rate pullback attract bounded subsets of the phase space under the evolution process is proved for the nonautonomous logistic equation and a system of reaction–diffusion equations with time-dependent external forces including the case of the FitzHugh–Nagumo system.

Global stability of nonautonomous logistic equations with a piecewise constant delay

Consider the following nonautonomous differential equation with a piecewise constant delay: { d N ( t ) d t = N ( t ) { r ( t ) − a ( t ) N ( t ) − b ( t ) N ( [ t ] ) } , n ≤ t < n + 1 , n = 0 , 1 , 2 , … , N ( 0 ) = N 0 > 0 , where r ( t ) is a nonnegative continuous function on [ 0 , + ∞ ) , r ( t ) ≢ 0 , a ( t ) ≥ 0 and b ( t ) > 0 . In this paper, using a recent result of a full affirmative answer to the Gopalsamy and Liu’s conjecture for the autonomous case of the above equation with a ( t ) ≡ a ≥ 0 and b ( t ) ≡ b > 0 , we derive three types of sufficient conditions for the global asymptotic stability of solution for the above nonautonomous logistic equation with a piecewise constant delay.

Survival analysis of stochastic single-species population models in polluted environments

This paper reports on the behaviors of single species models with and without pollution. We consider three basic models, one is deterministic, and others are stochastic. We first obtained the acute thresholds between local extinction and (stochastic) weakly persistent in the mean for population respectively. Then we study the attainability of population size 0 for the stochastic cases and show that a randomized non-autonomous logistic equation will be stochastic permanent under some conditions. Finally, we introduce some numerical simulink graphics to illustrate our main results.

On the population model of the non-autonomous logistic equation of second order with period-two parameters

In this paper we study the non-autonomous logistic difference equation with the nonnegative initial conditions x − 1, x 0. In particular, the periodic behavior, the attractivity of solutions and the stability of solutions are discussed in detail.

Effect of periodic environmental fluctuations on the Pearl–Verhulst model

We address the effect of periodic environmental fluctuations on the Pearl–Verhulst model in population dynamics and clarify several important issues very actively discussed in the recent papers by Lakshmi [Lakshmi BS. Oscillating population models. Chaos Solitons & Fractals 2003;16:183–6; Lakshmi BS. Population models with time dependent parameters. Chaos Solitons & Fractals 2005;26:719–21], Leach and Andriopoulos [Leach PGL, Andriopoulos K. An oscillatory population model. Chaos Solitons & Fractals 2004;22:1183–8], Swart and Murrell [Swart JH, Murrell HC. An oscillatory model revisited. Chaos Solitons & Fractals 2007;32:1325–7]. Firstly, we review general results regarding existence and properties of periodic solutions and examine existence of a unique positive asymptotically stable periodic solution of a non-autonomous logistic differential equation when r ( t ) > 0 . Proceeding to the case where r ( t ) is allowed to take on negative values, we consider a modified Pearl–Verhulst equation because, as emphasized by Hallam and Clark [Hallam TG, Clark CE. Non-autonomous logistic equations as models of populations in deteriorating environment. J Theor Biol 1981;93:303–11], use of the classic one leads to paradoxical biological conclusions. For a modified logistic equation with ω -periodic coefficients, we establish existence of a unique asymptotically stable positive periodic solution with the same period. Special attention is paid to important cases where time average of the intrinsic growth rate is non-positive. Results of computer simulation demonstrating advantages of a modified equation for modeling periodic environmental fluctuations are presented.

Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( t ) ] , where B ( t ) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that E [ 1 / N ( t ) ] has a unique positive T-periodic solution E [ 1 / N p ( t ) ] provided a ( t ) , b ( t ) and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 . We show that this equation is stochastically permanent and the solution N p ( t ) is globally attractive provided a ( t ) , b ( t ) and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and min t ∈ [ 0 , T ] a ( t ) > max t ∈ [ 0 , T ] α 2 ( t ) . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.

Global attractivity for discrete models of nonautonomous logistic equations

Consider the following discrete model of a nonautonomous logistic equation: { N ( n + 1 ) = N ( n ) exp { c ( n ) − ∑ j = 0 m b j ( n ) N ( n − j ) } , n ≥ 0 , N ( 0 ) = N 0 > 0 and N ( − j ) = N − j ≥ 0 , 1 ≤ j ≤ m , where c ( n ) and b j ( n ) , 0 ≤ j ≤ m , n ≥ 0 are bounded and c ( n ) > 0 , b 0 ( n ) > 0 , b j ( n ) ≥ 0 , 1 ≤ j ≤ m , n ≥ 0 . In this paper, using some kind of iterative method to the above equation, we establish sufficient conditions that ensure the global attractivity for solutions. The result is an extension of the former work [K. Uesugi, Y. Muroya, E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2004) 560–580] to the nonautonomous case.

Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations

The goal of this work is to study the forward dynamics of positivesolutions for the non-autonomous logistic equation $u_{t}-\Deltau=\lambda u-b(t)u^{p}$, with $p>1$, $b(t)>0$, for all $t\in\mathbb{R}$, $\lim_{t\to \infty }b(t)=0$. While the pullbackasymptotic behaviour for this equation is now well understood,several different possibilities are realized in the forwardasymptotic regime.

Global attractivity in nonautonomous logistic equations with delay

A new criterion is proposed for the global asymptotic stability of the positive periodic solutions to the following delay logistic equation: u ′ ( t ) = u ( t ) [ r ( t ) - a ( t ) u ( t ) - b ( t ) u ( t - τ ) ] . This result is preceded by the stability property of the zero solution to the linear equation x ′ ( t ) = - a ( t ) x ( t ) - b ( t ) x ( t - τ ) . Previous investigations are confirmed and generalized.

Error growth in the time-dependent logistic equation

In order to analyze the impact of time-dependent forcings in the behavior of the error growth we study a non-autonomous logistic equation. The sign of the Lyapunov exponent of the system depends on the frequency of the forcing. For finite initial errors, at initial stages the growth occurs in sub- and super-exponential ways, and finally oscillates around the saturation level. The form of the errors of the autonomous case is recovered when we average over many initial times, showing the importance of this parameter in time-dependent systems.

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

A non-autonomous Logistic type equation with infinite delay is investigated. For general nonautonomous case, sufficient conditions which guarantee the uniform persistence and globally attractivity of the system are obtained; For almost periodic case, by means of a suitable Lyapunov functional, sufficient conditions are derived for the existence and uniqueness of almost periodic solution of the system. Some new results are obtained.

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation d y ( t ) d t = − ∑ i = 0 m a i ( t ) g i ( y ( τ i ( t ) ) ) , t ≥ t 0 y ( t ) = φ ( t ) , t ≤ t 0 , where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that f ( 0 ) = 0 , 0 < g i ( x ) f ( x ) ≤ 1 , x ≠ 0 , 0 ≤ i ≤ m , and if f ( x ) ≢ x , then lim ⁡ x → − ∞ f ( x ) or lim ⁡ x → + ∞ f ( x ) is finite In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e x −1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.