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Models are presented suitable for a description of speciation processes arising due to reproductive isolation depending on genetic distance. The main attention is paid to the model of a one-dimensional closed population, which describes the evolution of littoral benthic organisms. In order to correspond the modeling results to the results obtained in the course of experimental phylogenetic studies, all individual-based models described here involve neutrally evolving and maternally inherited DNA sequence. Sub-samples of the resulting sequences were used for a posteriori phylogenetic inferences which then were compared to the "true" evolutionary histories.

Time evolutions of the circular cellular automata (CCA) are studied. The pattern seems to reach a dynamic balance state after several iterations in a fixed space. The time variations of population of each quadrant obey the famous logistic equation. It is found that although the initial patterns are very sensitive to the location of the first ring and the growth sequence, the dynamic patterns are independent of them. We propose that the CCA model may be used to simulate the evolution of biology.

A discrete logistic model is proposed for the Gross Domestic Product of an economy and its parameters are adjusted for some specific countries, taken from the Organization for Economic Co-operation and Development databases. Although this model fits “adequately” for some countries analyzed in this work, others show that it could be enhanced. A Caputo like fractional discrete model as a fractional version of the logistic one is presented, with the parameters stated previously. It is shown, by using the mean squared error, that this fractional version fits better for an economy whose behavior shows over time an isolated and unexpected boom period in its economy growth followed by a crisis period. For this type of dynamics, the fractional-order logistic equation improves the modeling given by an integer order logistic equation.

A logistic equation with piecewise constant arguments is investigated. Firstly, the linear stability of the model is studied. It is found that there exists a Hopf bifurcation when the parameter passes a critical value. Then the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution is derived by using the normal form method and center manifold theorem. Finally, computer simulations are performed to illustrate the analytical results found.

In this paper, we setup a model to analyze the structural change of an economic growth model with endogenous carrying capacity when the parameters vary. By integrating the logistically variable population carrying capacity into the classical Solow model, a two-dimensional dynamical system describing the model is obtained. It is proved that the dynamical system which describes the model has one, two or three equilibria under different conditions. It undergoes saddle-node bifurcations and the economy described by the model has multiple growth paths under the specified parameters. The "Malthusian trap" appears in the economy when the technological level and the saving rate are low or the carrying capacity grows slowly and the economy can escape the "Malthusian trap" by promoting technological levels or the saving rate or having high carrying capacity growth rate.

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the $X$, $Y$, and $Z$ state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The $X$ and $Z$ solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ($\mathrm{\text{sech}}$) and a hyperbolic secant squared function (${\mathrm{\text{sech}}}^2$), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the $Z$ component and the KdV equation were obtained. Additionally, the ODE for the $X$ component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.

In most models of pest control, the time of prevention and control is always assumed to be unrestricted. In practice, however, it is usually required in agriculture and forestry that we control the pest density to be lower than the economic injury level in a given time. By constructing the dynamics of logistic differential equation with impulsive effect, this paper describes the process of pest control through impulsive spraying pesticide in a given time, and then the existence condition of the solution to the boundary value problem of the dynamics is obtained, with which we solve the problem of pest control on a finite period of time and discuss the governing strategies.

The dynamic behavior of tri-trophic food chains consisting of resources, prey, predator and top-predator is dealt with. We compare a formulation whereby the prey growth is logistic, with a mass balance formulation. In the case of the mass balance formulation both the linear and the hyperbolic functional response are discussed. The consequences of the different formulations on the dynamics of a microbial food chain in chemostat situation are described. Bifurcation diagrams for the nonlinear dynamic systems are given. When the prey grows logistically there is no coexistence of the three species for biologically realistic parameter values for a microbial food chain. The same holds for the mass balance equations with a linear functional response for the prey. For a hyperbolic functional response, however, there is a stable equilibrium for the whole food chain in a rather large region of the parameter space. Furthermore, this model shows more complex dynamic behaviors; besides point attractors, limit cycles and chaotic attractors.

Power series, as an important means to analyze functions in different complex settings, are employed in various applied areas to solve differential equations and nonlinear problems and provide the assessment of intervals of convergence. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. The generalized Hukuhara difference and the generalized Hukuhara derivative are also used, and a power series representation is proposed for the solution of the fuzzy fractional logistic equation. Afterward, power series coefficients are obtained using a recursive formula. Finally, numerical experiments and illustrated results of the computations are presented to allow for more realistic decisions reflecting high complexity and underlying uncertainty. Thus, the numerical computations in our study reveal the effectiveness and accuracy of the power series method. Therefore, it is found that the fuzzy solution converges to the deterministic solution when uncertainty decreases, and, based on the technical analyses, it has been demonstrated that the results obtained are more fundamental in preventing geometric growth in nonlinear phenomena where uncertainties emerge due to impreciseness and inexactness.

Chaotic models as a rule are based on nonlinear recursive discrete equations, often illustrated by the logistic equation. In this paper, a conjecture is presented that gives an approximate solution to the continuous logistic equation. It is shown that if certain chaotic models are viewed as continuous rather than discrete, the strict borderline mathematical concept of the onset of chaos no longer retains its importance in some practical problems. In fact, it is more important to look at the degree of chaos, as illustrated in two medical problems dealing with blood flow and heart rate variability. The chaotic modeling approach is seen to be useful in analyzing experimental data.

Let f be a non-negative C¹-function on [0, ∞) such that f(u)/u is increasing and , where . Assume Ω ⊂ R^N is a smooth bounded domain, a is a real parameter and b ≥ 0 is a continuous function on , b≢0. We consider the problem Δu+au =b(x)f(u) in Ω and we prove a necessary and sufficient condition for the existence of positive solutions that blow-up at the boundary. We also deduce several existence and uniqueness results for a related problem, subject to homogeneous Dirichlet, Neumann or Robin boundary condition.

The nonautonomous single-species Kolmogorov system is studied in this paper. Average conditions are obtained for permanence, global attractivity and extinction in the system. Applications of our main results to logistic equation and generalized logistic equation are given. It is shown that our average conditions are improvement of those in Vance and Coddington [J. Math. Biol.27 (1989) 491–506] and some published literature on the system.

A large number of studies have examined issues of dividend policy, while they rarely consider the investment decision and dividend policy jointly from a non-steady state to a steady state. We extend Higgins’ (1977, 1981, 2008) sustainable growth rate model and develop a dynamic model which jointly optimizes the growth rate and payout ratio. We optimize the firm value to obtain the optimal growth rate in terms of a logistic equation and find that the steady-state growth rate can be used as the benchmark for the mean-reverting process of the optimal growth rate. We also investigate the specification error of the mean and variance of dividend per share when introducing the stochastic growth rate. Empirical results support the mean-reverting process of the growth rate and the importance of covariance between the profitability and the growth rate in determining dividend payout policy. In addition, the intertemporal behavior of the covariance may shed some light on the fact of disappearing dividends over decades.

A large number of studies have examined issues of dividend policy, while they rarely consider the investment decision and dividend policy jointly from a non-steady state to a steady state. We extend Higgins’ (1977, 1981, and 2008) sustainable growth rate model and develops a dynamic model which jointly optimizes the growth rate and payout ratio. We optimize the firm value to obtain the optimal growth rate in terms of a logistic equation and find that the steady state growth rate can be used as the benchmark for the mean-reverting process of the optimal growth rate. We also investigate the specification error of the mean and variance of dividend per share when introducing the stochastic growth rate. Empirical results support the mean-reverting process of the growth rate and the importance of covariance between the profitability and the growth rate in determining dividend payout policy. In addition, the intertemporal behavior of the covariance may shed some light on the fact of disappearing dividends over decades.

A general n-node network is considered for which, in absence of interactions, each node is governed by a logistic equation. Interactions among the nodes take place in the form of competition, which also includes adaptive abilities through a (short term) memory effect. As a consequence the dynamics of the network is governed by a system of n² nonlinear ordinary differential equations. As a first step, equilibria and their stability are investigated analytically for the general network in dependence of the relevant parameters, namely the strength of competition, the adaptation rate and the network size. The existence of classes of invariant subspaces, related to symmetries, allows the introduction of a reduced model, four dimensional, where n appears as a parameter, which give full account of existence and stability for the equilibria in the network.

The Beverton–Holt equation, usually treated as a rational difference equation, is shown in fact to be a logistic difference equation. Based on a crucial transformation connected to logistic equations, an elementary proof of the Cushing–Henson conjectures is given.

The field of environmental sciences is abundant with various interfaces and is the right place for the application of new fundamental approaches leading towards a better understanding of environmental phenomena. For example, following the definition of environmental interface by Mihailovic and Balaž [23], such interface can be placed between: human or animal bodies and surrounding air, aquatic species and water and air around them, and natural or artificially built surfaces (vegetation, ice, snow, barren soil, water, urban communities) and the atmosphere. Complex environmental interface systems are open and hierarchically organised, interactions between their constituent parts are nonlinear, and the interaction with the surrounding environment is noisy. These systems are therefore very sensitive to initial conditions, deterministic external perturbations and random fluctuations always present in nature. The study of noisy non-equilibrium processes is fundamental for modelling the dynamics of environmental interface systems and for understanding the mechanisms of spatio-temporal pattern formation in contemporary environmental sciences, particularly in environmental fluid mechanics. In modelling complex biophysical systems one of the main tasks is to successfully create an operative interface with the external environment. It should provide a robust and prompt translation of the vast diversity of external physical and/or chemical changes into a set of signals, which are “understandable” for an organism. Although the establishment of organisation in any system is of crucial importance for its functioning, it should not be forgotten that in biophysical systems we deal with real-life problems where a number of other conditions should be reached in order to put the system to work. One of them is the proper supply of the system by the energy. Therefore, we will investigate an aspect of dynamics of energy flow based on the energy balance equation. The energy as well as the exchange of biological, chemical and other physical quantities between interacting environmental interfaces can be represented by coupled maps. In this chapter we will address only two illustrative issues important for the modelling of interacting environmental interfaces regarded as complex systems. These are (i) use of algebra for modelling the autonomous establishment of local hierarchies in biophysical systems and (ii) numerical investigation of coupled maps representing exchange of energy, chemical and other relevant biophysical quantities between biophysical entities in their surrounding environment.

The goal of the paper is chaos examination in multiplicative systems. The paper collects results of numerical simulations as well as presentation of methods applicable in the case of multiplicative systems. Chaos examination concerns one-dimensional multiplicative version of logistic equation and multi-dimensional nonlinear system described with multiplicative derivatives. The classical Lorenz system transformed into multiplicative version was chosen for analysis of stability and chaotic behaviour.

A geometric method is presented for studying the zeros of the Riccati equation and some applications to bio-economical prognoses are given.