Dynamical Behavior of Logistic Maps

Open Journal of Modelling and Simulation, 2014

In this paper, some theoretical mathematical aspects of the known predator-prey problem are considered by relaxing the assumptions that interaction of a predation leads to little or no effect on growth of the prey population and the prey growth rate parameter is a positive valued function of time. The predator growth model is derived considering that the prey follows a known growth models viz., Logistic and Von Bertalanffy. The result shows that the predator's population growth models look to be new functions. For either models, the predator population size either converges to a finite positive limit or to 0 or diverges to +∞. It is shown algebraically and illustrated pictorially that there is a condition at which the predator-prey population models both converge to the same finite limit. Derivations and simulation studies are provided in the paper. Analysis of equilibrium points and stability is also included.

Proceedings of the International Conference on Information Technology Interfaces, ITI, 2009

We present new populational growth models, generalized logistic models which are proportional to beta densities with shape parameters p and 2, where p > 1, with Malthusian parameter r. The complex dynamical behaviour of these models is investigated in the parameter space (r, p), in terms of topological entropy, using explicit methods, when the Malthusian parameter r increases. This parameter space is split into different regions, according to the chaotic behaviour of the models.

Bulletin of Mathematical Biology, 1980

Current research into the dynamics of iterative ecological and biological models has lead to a number of theorems concerning the existence of various types of iterative dynamical behavior. In particular, much study has been done on the dynamical behavior of the "simplest dynamical system" fb(x)=bx(1--x), which is just the canonical discrete form of logistic growth equations found in ecology, sociobiology, and population biology. In this paper, we make use of some of the techniques and concepts of topological dynamics to construct a number of generalized conjugacy theorems. These theorems are then used to demonstrate that the mapping fb has a number of conjugacy classes in which the dynamics of the iterates is equivalent to within a change of variables. The concepts of fitness and survival in logistic equations are then shown to be independent, if we follow certain intuitive definitions for these concepts. This conclusion follows from a comparison of the conjugacy classes of the function fb and the extinction sets of ~.

2017

In an attempt to discover the effect of recurrence on the topological dynamics, a nonlinear function whose regime of periodicity amounts to recurrence was considered, thus the logistic function. This research seeks to study the logistic function as to how it really behaves. In the field of dynamics most especially this function in terms of discrete form has been studied. Logistic equation as a model based on population growth was initially originated by the famous Pierre-Francios Verhulst. It is a continuous form written as dx dt = r(x − x2), which depend on time. It can be restructured from the continuous form into a discrete form known as logistic function, written as; xn+1 = rxn(1 − xn), with n = 0,1,2,3 ... ... ,xn is 1 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. E-mail: nanaamonoo12@yahoo.com 2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. E-mail: obengdentehw@yahoo.com 3 Departmen...

In an attempt to discover the effect of recurrence on the topological dynamics, a nonlinear function whose regime of periodicity amounts to recurrence was considered, thus the logistic function. This research seeks to study the logistic function as to how it really behaves. In the field of dynamics most especially this function in terms of discrete form has been studied. Logistic equation as a model based on population growth was initially originated by the famous Pierre-Francios Verhulst. It is a continuous form written as í µí±‘í µí±¥ í µí±‘í µí±¡ = í µí±Ÿ(í µí±¥ − í µí±¥ 2), which depend on time. It can be restructured from the continuous form into a discrete form known as logistic function, written as; í µí±¥ í µí±›+1 = í µí±Ÿí µí±¥ í µí±› (1 − í µí±¥ í µí±›), with í µí±› = 0,1,2,3 … … ,í µí±¥ í µí±› is

Applied Mathematical Modelling, 2007

In this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay s passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions.

Differential Equations & Applications

In this paper we interpret the global stability properties of the delayed single species chemostat in terms of monotone dynamics on an asymptotically invariant hyperplane in the state space. The consequence is a translation of advanced analysis and delay differential equations into sign checks and ordinary differential equations for an important single species model with explicit resource dynamics. Complete proofs are included, since the limiting behavior at asymptotically invariant sets may not agree with the limiting behavior of the original system even in the finite dimensional case (Thieme (1992)).

Journal of Mathematical Biology, 1995

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.

On infinitesimally short time interval various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann (1986)). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to dependency terms affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open.