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Dynamics of a plant–herbivore model with differential–difference equations

Abstract: This paper studies the behavior of a plant–herbivore model including both differential and difference equations. To analyze global behavior of the model, we consider the solution of the system in a certain subinterval which gives to system of difference equations. The boundedness characters, the periodic nature, both local and global stability conditions of the plant–herbivore system are investigated. Numerical studies indicate that the system exhibits Neimark–Sacker bifurcation for different parameter values in certain regions.

ABOUT THE AUTHOR Senol Kartal is an associate professor in Nevsehir Haci Bektas Veli University, Turkey. His PhD thesis is about the population dynamics (Modeling of Tumor Immune System Dynamics Using System of Difference Equations and Its Stability Analysis). His research interests are related to population dynamics, discrete and continuous dynamical system, and bifurcation theory. He has published his research contributions in some internationally renowned journals whose publishers are Elsevier, Taylor & Francis, Wiley, and other journals.

PUBLIC INTEREST STATEMENT
In this study, we consider a plant-herbivore model which consists of ordinary differential equations. Our aim is to build a better understanding of how both discrete and continuous times affect the dynamic behavior of plant-herbivore interactions. Therefore, we add discrete time to this model and obtain a system of differential equations with piecewise constant arguments which gives system of difference equations. The boundedness characters, the periodic nature, both local and global stability conditions of the system are investigated.
It is well known that discrete time models governed by difference equations are more appropriate than the continuous time models when the populations have non-overlapping generations. So, a significant number of the study on the mathematical models of plant-herbivore interactions are described by the system of difference equations (Agiza et al., 2009;Danca et al., 1997;Mukherjee et al., 2011;Sui et al., 2007). In addition, working with difference equations instead of differential equations allows us to some advantages. Discrete dynamical models can bring about easier computational methods for the persistence, periodic solutions, boundedness, local and global properties of the dynamical system.
In plant-herbivore interactions, delay differential equations may widely occur due to herbivore damage and deployment of inducible defenses (Das & Sarkar, 2001;Ortega-Cejas et al., 2004;Sun et al., 2014). From this point of view, Sun et al. (2014) and et all have constructed a reaction-diffusion model with delay governed by system of partial differential equations where the effect of time delay on the herbivore cycles is investigated. In addition, the properties of delay differential equations are very close to differential equation with piecewise constant arguments. In Cooke and Györi study (1994), it was pointed out that these equations can be used to get approximate solutions to delay differential equations that include discrete delays. In such biological situations, dynamics of growth and death of populations can be described by differential equations otherwise, difference equations may reflect the interaction of two populations such as competition or predation phenomena (Gurcan, Kartal, Ozturk, & Bozkurt, 2014;Kartal & Gurcan, 2015). In the literature, various types of biological model consisting of differential equations with piecewise constant arguments have been analyzed using the method of reduction to discrete equations (Busenberg & Cooke, 1982;Gopalsamy & Liu, 1998;Gurcan et al., 2014;Kartal & Gurcan, 2015;Liu & Gopalsamy, 1999;Öztürk, Bozkurt, & Gurcan, 2012).
In the present paper, our aim is to build a better understanding of how both discrete and continuous times affect the dynamic behavior of plant-herbivore interactions. So we will reconsider the model (see Chattopadhayay et al., 2001) as a system of differential equations with piecewise constant arguments such as which include both differential and difference equations. In this model, x(t) and y(t) represent the density of plant and herbivore population, respectively, [[t]] denotes the integer part of t ∈ [0, ∞) and all these parameters are positive. The parameter r, K, and is the intrinsic growth rate, environmental carrying capacity, and specific predation rate of plant species, respectively. s represents the death rate of herbivores and is the conversion factor of herbivores (Chattopadhayay et al., 2001).

(t)y([[t]]) represent the loss of plant population and x([[t]])y(t)
is conversion factor of herbivores which include both discrete and continuous time for a each populations. So the plant-herbivore interaction is considered in a certain subinterval and is modeled using a system of differential equations with piecewise constant arguments.

Local and global stability analysis
System (1.2) can be written an interval t ∈ [n, n + 1) as follows: By solving each equations of the system (2.1) and letting t → n + 1, we obtain a system of difference equations System (2.2) reflects the dynamical behavior of the system of differential equations with piecewise constant arguments. So we will consider the system of difference equation to analyze the global behavior of system (1.2).
The equilibrium points of system (2.2) can be obtained as We note that the positive equilibrium of the system exists if > ks. Now, we will find Jacobian matrix of the system to investigate the dynamic behavior of the model.
Theorem 2.1. The equilibrium points E 0 and E 1 are saddle point.
Proof At the equilibrium point E 0 , the Jacobian matrix is the form The matrix J 0 has eigenvalues 1 = e r , 2 = e −s . Hence 1 > 1 and 2 < 1 and consequently E 0 is saddle point. On the other hand, the Jacobian matrix J 1 at the point E 1 is which gives eigenvalues 1 = e −r and 2 = e −s+ k . Considering the condition > ks, we can say that E 1 is saddle point.
On the other hand, the Jacobian matrix J * at the positive equilibrium point E * is which yields the following characteristic equation Now, we can apply Schur-Cohn criterion to determine stability conditions of the system with characteristic equation p( ).

Theorem 2.4 The system has no prime period-two solutions.
Proof On the contrary, suppose that the system (2.2) has a distinctive prime period-two solutions where w 1 ≠ w 2 , and q 1 ≠ q 2 , and w i , q i are positive real numbers for i ∈ {1, 2}. Then, from system (2.2) one has Since q 1 ≠ q 2 , we have w 2 − s ≠ 0 and w 1 − s ≠ 0. From the second and last equation in the system, we have If q 2 is written the above equation, we hold This equation must satisfy which is a contradiction w 2 − s ≠ 0 and w 1 − s ≠ 0.
Theorem 2.5 Let A 1 = r − y(n) and A 2 = x(n) − s. Suppose that the conditions of Theorem 2.1 hold and Then the positive equilibrium point of system (2.2) is global asymptotically stable.
Proof We define a Lyapunov function as where q = (x,ȳ) is positive equilibrium point of system (2.2).

Bifurcation analysis
In this section, we investigate existence of stationary bifurcation (fold, transcritical, and pitchfork bifurcation), period doubling bifurcation, and Neimark-Sacker bifurcation for the system (2.2). All of these bifurcations can be analyzed under the set of algebraic conditions that is called Schur-Cohn criterion. It is well known that the system may undergo stationary bifurcation if and only if p(1) = 0, p(−1) > 0, D + 1 > 0 and D − 1 > 0. On the other hand, inequalities p(1) > 0, p(−1) = 0, D + 1 > 0 and D − 1 > 0 give the conditions of period doubling bifurcation. But considering (2.3) and (2.4), it is easily seen that these conditions do not hold for the system. Therefore, stationary bifurcation and period doubling bifurcation do not exist for the system. Now, we can investigate the existence of Neimark-Sacker bifurcation for the plant-herbivore model (Hone, Irle, & Thurura, 2010). The algebraic condition of Neimark-Sacker bifurcation can be obtained from the analysis of inequalities p(1) > 0, p(−1) > 0, D + 1 > 0 and D − 1 = 0. In local stability analysis, we have already shown that the inequalities p(1) > 0, p(−1) > 0, D + 1 > 0 are always exist. Therefore, we will only analyse the equation D − 1 = 0 to determine Neimark-Sacker bifurcation condition.

Theorem 3.1 System (2.2) undergoes Neimark-Sacker bifurcation if and only if
Proof This result comes from the analysis of D − 1 = 0.
Using the condition of Theorem 3.1 with the parameters given in Figure 1, we have the Neimark-Sacker bifurcation point as K = 102 ( Figure 2).

Result and discussion
In this paper, dynamics of a discrete-continuous time plant-herbivore model has been investigated. Local and global stability properties of the positive equilibrium point are analyzed. It is interesting to note that when conversion factor of herbivores becomes low then the system converges to a stable situation. On the other hand, we investigate possible bifurcation types for the system and observe that the system exhibits Neimark-Sacker bifurcation. This type of bifurcation has been observed in many plant-herbivore models (Liu et al., 2008;Saha & Bandyopadhyay, 2005;Zhao et al., 2015) and shows that periodic or quasi-periodic solutions occur as a result of a limit cycle.