Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 756315, 13 pages

http://dx.doi.org/10.1155/2015/756315

## On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

Received 11 October 2014; Accepted 5 January 2015

Academic Editor: Gualberto Solís-Perales

Copyright © 2015 Mehbuba Rehim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models.

#### 1. Introduction

Phytoplankton are one of the most important components of the marine ecosystem. They not only form a basis for all aquatic food chains but also perform a very useful service by producing a huge amount of oxygen for human and other living animals after absorbing carbon-dioxide from surrounding environments [1]. Zooplankton are microscopic animals that eat other plankton and serve as a most favorable food source for fish and other aquatic animals. During the recent years, many authors have studied the system between zooplankton and phytoplankton. Authors in [2] have dealt with a nutrient-plankton model in an aquatic environment in the context of phytoplankton bloom. In [3] the effect of seasonality and periodicity on plankton dynamics is investigated. In [4], two plankton ecosystem models with explicit representation of viruses and virally infected phytoplankton are presented.

The most common features of the phytoplankton population is rapid increase of biomass due to rapid cell proliferation and almost equally rapid decrease in populations, separated by some fixed time period. This type of rapid change in phytoplankton population density is known as “*bloom*” [5]. Due to the accumulation of high biomass or to the presence of toxicity, some of these blooms, more adequately called “*harmful algal blooms*” [6], are noxious to marine ecosystems or to human health and can produce great socioeconomic damage. There has been a global increase in harmful plankton blooms in last two decades [7–9].

Because of the difficulty in measuring plankton biomass, mathematical modeling of plankton population is an important alternative method of improving our knowledge of the physical and biological processes relating to plankton ecology [10–15]. In [16], nutrient-plankton-zooplankton interaction models with a toxic substance which inhibits either the growth of phytoplankton, zooplankton, or both trophic levels are proposed and studied. In [17], the authors have constructed a mathematical model for describing the interaction between a nontoxic and toxic phytoplankton with a single nutrient.

Based on the fact that the release of toxin from phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton may die after some time lapse of the bloom of toxic phytoplankton (see http://www.mote.org/, http://www.mdsg.umd.edu/), models incorporating time delay in diverse biological models are extensively reviewed by Beretta and Kuang [18], Gopalsamy [19], Cooke and Grossman [20], and Cushing [21]. The discrete time delay has potential to change the qualitative behavior of the dynamical systems [22–29]. Chattopadhyay et al. proposed a delay model incorporating time lag in toxin liberation by phytoplankton to avoid predation by zooplankton [28]. In [28], the authors introduced distribution delay and discrete to toxin liberation term. Due to discrete time delay in toxin liberation, the local existence of periodic solution through Hopf bifurcation has been obtained in [28].

In [5, 10, 28–31], the following plausible toxic-phytoplankton-zooplankton system has been studied: where is the density of the toxin-producing phytoplankton (TPP) population and is the density of zooplankton population at any instant of time . In model (1), is the intrinsic growth rate and is the environmental carrying capacity of TPP population. The term describes the functional response for the grazing of phytoplankton by zooplankton, and is the maximum uptake rate for zooplankton. denotes the ratio of biomass conversion and is the natural death rate of zooplankton. The function represents the distribution of toxic substance which ultimately contributes to the death of zooplankton populations, and the parameter denotes the rate of toxic substances produced by per unit biomass of phytoplankton.

Model (1) has been studied for the following cases: when is linear but is Holling type II [10, 28, 29] or Holling type III [10]; when and are both Holling type II [5, 10] or Holling type III [10]; when is Holling type II while is Holling type III [10].

In above cases, and are both increasing functions of over the entire interval . However, in some cases, very high substrate concentrations in the lakes actually inhabit the growth of phytoplankton cells. Moreover, with the substrate concentrations increasing unlimitedly, some kind of microorganism will die eventually [32]. To describe the above phenomenon accurately, we consider from a different point of view. Adopting the idea used in [32], we assume that there exists a constant such that is increasing over the interval and is decreasing on the interval . More precisely, we use the so-called Tissiet functional response of the form of (see, e.g., [32]). This type of functional response takes care of the fact that the predation rate decreases at large toxic-phytoplankton density.

The remainder of this paper is organized as follows. The model is described in Section 2. In Section 3, we state and prove the positivity and boundedness of the solutions. Then, in Section 4, equilibria, their existence, and local asymptotic stability are considered. Under the aids of numerical simulation method, we further analyse the model and determine if there is a parameter range for the delay parameter , where more complicated dynamics occur. In Section 5, the permanence of the system is discussed by some analytic techniques on limit sets of differential dynamical systems. Finally, a brief discussion is presented in Section 6.

#### 2. State of the Model

In a real ecological context, the interaction between phytoplankton and zooplankton will not be essentially instantaneous. Instead, the response of zooplankton to contacts with phytoplankton is likely to be delayed due to a gestation period. Another fact, during the interaction between phytoplankton and zooplankton, is that the liberation of toxic substances by phytoplankton must be mediated by some time lag which is required for the maturity of toxic-phytoplankton. Let stand for the time delay in conversion of food to viable biomass for the species, and is the discrete time period required for the maturity of phytoplankton cells to liberate toxic substances. Based on model 3 in [5], we intend to study a model system with the assumption that and are described by same type of function, namely, Tissiet functional response. Then we arrive at the following model: where the parameter is the intrinsic growth rate and is the environmental carrying capacity of TPP population. The constant is the maximum per capita grazing rate, denotes the ratio of biomass conversion, denotes the rate of toxic substances produced by per unit biomass, is the natural death rate of the zooplankton. All the parameters in system (2) are positive constants with their usual ecological meanings.

Here we observe that if there is no delay (i.e., ) and , then . Hence, throughout our analysis, we assume that .

For the sake of simplicity of mathematical analysis, in this paper, we consider model (2) for the special case: the time delay in conversion of food to viable biomass for the species equal to the discrete time period required for the maturity of phytoplankton cells to liberate toxic substances; that is, .

By performing the following scaling for model (2): we obtain a dimensionless system in the state variables , , which can be written, by removing the stars, in the form with initial conditions: where .

#### 3. Positive and Boundedness

In this section, we consider the positive and boundedness of the solutions of model (4) with initial condition (5). One has the following theorem.

Theorem 1. *Let , on and , . Then*(a)*all the solutions of (4) with initial condition (5) exist on for some constant and are unique and positive for ,*(b)*, , where ,*(c)*if , then , where . Further, the subset
**is positively invariant with respect to (4).*

*Proof. *By Theorems 2.1 and 2.3 in Hale and Lunel [33], solutions of (4) with initial data exist on for some and are unique. Suppose is a solution of (4) for . Without loss of generality, assume that is the maximum internal of the solution and if the solution exists for any . Integrating the first equation of (4) gives
To prove the for any , use the method of contradiction. Suppose there exists a such that
From the second equation of system (4), we have
This contradicts . Hence, for all . This completes the proof of conclusion (a) in Theorem 1.

It follows from the first equation of (4) that , which implies that . Define , . Then from (4) we obtain
Applying the theorem of differential inequality we obtain that
Therefore, . Thus, there is a constant , such that for all . This completes the proof of conclusion (b) in Theorem 1.

From the first equation of system (4) we get
which implies that .

For any , let be the solution of (4) with the initial function . If there is a such that , then for some and . Hence, it follows from the first equation of (4) that
which is a contradiction to . So, for all .

It is easy to prove that if there is a such that , then for all one has . Which implies that for all . This completes the proof of conclusion (c) in Theorem 1.

*Remark*. The conclusions (b) and (c) in Theorem 1 indicate that if the ratio of biomass conversion is less than certain value, then phytoplankton population will be persistent.

#### 4. Equilibrium, Stability, and Hopf Bifurcation

##### 4.1. Existence of Equilibria

It is easy to see that model (4) has two boundary equilibria and . To discuss the existence of the positive equilibria, we work on the equation Denote the left-hand side of the second equation in (14): Then , , Let , we have , where From this we know that the function is monotone increasing in the interval and monotone decreasing in the interval . Hence, reaches its maximum on the interval . Therefore, we have the following results.

Theorem 2. *The following statements hold.*(1)*If , then equation has no roots in the interval (Figure 1(a)). In this case, for system (4), there is no positive equilibrium.*(2)*If and , then equation has only one root, , in the interval , where (Figure 1(c)). In this case, for system (4), there exists a single positive equilibrium .*(3)*If and , then equation has two distinct roots, and , in the interval , where (Figure 1(d)). In this case, for system (4), there are two distinct positive equilibria and .*(4)*If , then equation has a unique root, , in the interval , (Figure 1(b)). In this case, for system (4), there is a unique positive equilibrium .*(5)*If and , then equation has two distinct roots, and , in the interval , where (Figure 1(e)). In this case, for system (4), there is also unique positive equilibrium ,**where , , are determined by the first equation in (14).*