Papers by Senada Kalabusic
Computational & Applied Mathematics, Oct 23, 2023
Qualitative Theory of Dynamical Systems, Jan 29, 2022
Journal of Difference Equations and Applications, Feb 23, 2022
Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behav... more Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:10.1007/s12190-021-01503-2], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number We establish the global asymptotic stability of the disease-free equilibrium (GAS) for We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for . For the transcritical bifurcation appears. For we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number .
Qualitative Theory of Dynamical Systems, Jun 17, 2020
In this paper, by using the analytical approach, we investigate the global behavior and bifurcati... more In this paper, by using the analytical approach, we investigate the global behavior and bifurcation in a class of host-parasitoid models when a constant number of the hosts are safe from parasitism. We find the conditions for the existence and stability of the equilibria. We detect the existence of the Neimark-Sacker bifurcation under certain conditions. We explicitly derived the approximation of the limit curve depending on the parameters that appear in the model. We show that a locally asymptotically stable equilibrium can never be transformed into unstable by increasing a constant number of hosts that are using a refuge. Specially, we consider the effect of constant host refuge in (S), (HV), and (PP) models.The obtained results show that the constant number of hosts in refuge affects the qualitative behavior of these models in comparison to the same models without refuge. The theory is confirmed and illustrated numerically.
Springer proceedings in mathematics & statistics, 2023
Novi Sad Journal of Mathematics, 2003
Advances in Difference Equations, Jan 7, 2021
This paper is motivated by the series of research papers that consider parasitoids' external inpu... more This paper is motivated by the series of research papers that consider parasitoids' external input upon the host-parasitoid interactions. We explore a class of host-parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark-Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host-parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.
International Journal of Biomathematics, Apr 26, 2023
This paper deals with the host–parasitoid model, where the logistic equation governs the host pop... more This paper deals with the host–parasitoid model, where the logistic equation governs the host population growth, and a proportion of the host population can find refuge. The equilibrium points’ existence, number, and local character are discussed. Taking the parameter regulating the parasitoid’s growth as a bifurcation parameter, we prove that Neimark–Sacker and period-doubling bifurcations occur. Despite the complex behavior, it can be proved that the system is permanent, ensuring the long-term survival of both populations. Furthermore, it was observed that the presence of the proportional refuge does not significantly influence the system’s behavior compared to the system without a proportional refuge.
We investigate the local stability and the global asymptotic stability of the following two diffe... more We investigate the local stability and the global asymptotic stability of the following two difference equation xn+1 = βxnxn−1 + γxn−1 Ax 2 n + Bxnxn−1 , x0 + x−1 > 0, A + B > 0 xn+1 = αx 2 n + βxnxn−1 + γxn−1 Ax 2 n , x0 > 0, A > 0 where all parameters and initial conditions are positive.
We give some general results about global dynamics of an anti-competitive system of the form {xn+... more We give some general results about global dynamics of an anti-competitive system of the form {xn+1 = T1(xn; yn) yn+1 = T2(xn; yn) ; n = 0; 1; 2; : : : where T1 : L × J → L, T2: L ×J → J and (x0; y0) 2 I∈ J ; and functions T1 and T2 are continuous and T1(x; y) is non-increasing in x and non- decreasing in y while T2(x; y) is non-decreasing in x and non-increasing in y. We illustrate our results by means of an example which shows a variety of typical dynamical behavior for an anti-competitive system. Copyright © 2013 Watam Press
Journal of Biological Systems, Jun 5, 2023
This paper studies the dynamics of a class of host-parasitoid models with host refuge and the str... more This paper studies the dynamics of a class of host-parasitoid models with host refuge and the strong Allee effect upon the host population. Without the parasitoid population, the Beverton–Holt equation governs the host population. The general probability function describes the portion of the hosts that are safe from parasitism. The existence and local behavior of solutions around the equilibrium points are discussed. We conclude that the extinction equilibrium will always have its basin of attraction which implies that the addition of the host refuge will not save populations from extinction. By taking the host intrinsic growth rate as the bifurcation parameter, the existence of the Neimark–Sacker bifurcation can be shown. Finally, we present numerical simulations to support our theoretical findings.
Nonlinear Studies, Mar 31, 2011
ABSTRACT We prove fixed point theorems for monotone mappings in partially ordered complete metric... more ABSTRACT We prove fixed point theorems for monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition for all points that are related by a given ordering. We also give a global attractivity result for all solutions of the difference equation z n+1 =F(z n ,z n-1 ),n=2,3,⋯ where F satisfies certain monotonicity conditions with respect to the given ordering.
ABSTRACT We extend the known results of the nonautonomous difference equation in the title to the... more ABSTRACT We extend the known results of the nonautonomous difference equation in the title to the situation where (i) the parameters beta(n) and gamma(n) are period-two sequences of nonnegative real numbers with gamma(n) not identically zero; (ii) the parameters An and B(n) are period-two sequences of positive real numbers; and (iii) the initial conditions x(-1) and x(0) are such that x(-1), x(0) is an element of [0, infinity) and x(-1) + x(0) is an element of (0, infinity).
Mathematics, Jan 12, 2018
We investigate global dynamics of the following second order rational difference equation x n+1 =... more We investigate global dynamics of the following second order rational difference equation x n+1 = x n x n−1 +αx n +βx n−1 ax n x n−1 +bx n−1 , where the parameters α, β, a, b are positive real numbers and initial conditions x −1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.
Journal of Difference Equations and Applications, Aug 1, 2003
We investigate the global character of solutions of the equation in the title with positive param... more We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.
Discrete and Continuous Dynamical Systems - B
International Journal of Biomathematics
Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasi... more Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.
Advances in Difference Equations, 2021
This paper is motivated by the series of research papers that consider parasitoids’ external inpu... more This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level s...
Qualitative Theory of Dynamical Systems, 2020
In this paper, by using the analytical approach, we investigate the global behavior and bifurcati... more In this paper, by using the analytical approach, we investigate the global behavior and bifurcation in a class of host-parasitoid models when a constant number of the hosts are safe from parasitism. We find the conditions for the existence and stability of the equilibria. We detect the existence of the Neimark-Sacker bifurcation under certain conditions. We explicitly derived the approximation of the limit curve depending on the parameters that appear in the model. We show that a locally asymptotically stable equilibrium can never be transformed into unstable by increasing a constant number of hosts that are using a refuge. Specially, we consider the effect of constant host refuge in (S), (HV), and (PP) models.The obtained results show that the constant number of hosts in refuge affects the qualitative behavior of these models in comparison to the same models without refuge. The theory is confirmed and illustrated numerically.
Mathematics, 2018
We investigate global dynamics of the following second order rational difference equation x n+1 =... more We investigate global dynamics of the following second order rational difference equation x n+1 = x n x n−1 +αx n +βx n−1 ax n x n−1 +bx n−1 , where the parameters α, β, a, b are positive real numbers and initial conditions x −1 and x 0 are arbitrary positive real numbers. The map associated to the right-hand side of this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the corresponding parametric space. In most cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability.
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Papers by Senada Kalabusic