A nutrient-phytoplankton-zooplankton model is established, where harvest effort on phytoplankton population and seasonal intrinsic growth of nutrient biomass are considered. Positivity and boundedness of solutions of model system are studied. By analyzing the characteristic equation of linearized system corresponding to the proposed model system, local asymptotic stability around interior equilibrium is studied, which reveals that interior equilibrium loses its stability at some critical value of predation rate. By utilizing Poincaré surface of section and computation of Lyapunov exponents, numerical experiments are performed to discuss the complex dynamical behavior of model system. Model system shows rich dynamic behavior including limit cycle and chaotic attractor due to variation of predation rate and seasonal intrinsic growth rate. Further numerical experiments show that reasonable harvesting has stabilizing effect on population dynamics that prevents chaotic behavior. Numerical simulations are carried out to show consistency with theoretical analysis obtained in this paper.
Plankton are microscopic organisms that float freely within marine system. They are made up of tiny plants (called phytoplankton) and tiny animals (called zooplankton). Phytoplankton population is the dominant primary producer in the marine system, which produces approximately forty percent oxygen through photosynthesis. By primary production, death, and sinking, they exert a global scale influence on climate by effectively transporting
The dynamics of marine system have long been one of the dominant themes in mathematical biology. Marine population models are generally based on compartmental models consisting of many coupled nonlinear differential equations. Some of these models seek to contain as many as physical and biological mechanisms. In recent years, there is a growing explicit biological physiological body of evidence [
One of the most interesting topics in mathematical ecology concerns the seasonal intrinsic growth of nutrient biomass in marine ecosystem. Countless organisms live in seasonal marine environment, and many parameters of such system may oscillate simultaneously and not necessarily in plane. It is pertinent to note that intrinsic growth of nutrient biomass is usually periodically adjusted, which may be caused by periodic variation of fecundity of nutrient within marine ecosystem. Recently, the model systems of marine ecosystem with periodic external force have gathered new attention because of their complex dynamical behavior, especially chaotic behavior. A variety of studies have been performed on the interactions between seasonality and internal biological rhythms, whose dynamical effects on nutrient-phytoplankton ecosystem are discussed in [
The organization of this paper is as follows. By incorporating harvest effort on phytoplankton population and seasonal intrinsic growth of nutrient biomass, we extend the model proposed in [
The classical Rosenzweig-MacArthur prey-predator model [
In the following part, some hypotheses are proposed, which are utilized to construct the mathematical model and discuss population dynamics of a harvested nutrient-phytoplankton-zooplankton ecosystem with seasonality. It is assumed that the phytoplankton population feeds on nutrient and the zooplankton population grazes on the phytoplankton for survival. Zooplankton population is solely dependent upon phytoplankton as their most favorable food source and a variation in phytoplankton population density has a great impact on the growth of zooplankton population. In this paper, we will consider a three-tier model of nutrient-phytoplankton-zooplankton with the usage of model system ( Due to its extensive utilization in the field of food and hygienic field, phytoplankton population is extensively exploited in the real world. In this paper, we will consider the harvest effort on phytoplankton population. A scalar It is observed that nutrient biomass is recycled within nutrient-phytoplankton-zooplankton ecosystem [ It is assumed that the intrinsic growth of nutrient biomass varies periodically, which may be caused by periodic variation of fecundity of nutrient within marine ecosystem. Consequently, the intrinsic growth rate of nutrient biomass
Based on model system (
All solutions of model system (
The model system (
Due to the lemma in [
Solutions of model system (
Let
According to Theorem
Based on the above analysis, it can be obtained that
In this section, qualitative analysis of model system (
In the case of
In order to reduce number of parameters and facilitate dynamical analysis of model system, model system (
Since the interior equilibrium biologically relates to simultaneous survival of nutrient biomass, phytoplankton population, and zooplankton population, we will mainly concentrate on dynamical analysis of model system around the interior equilibrium in this paper.
According to model system (
Based on Routh-Hurwitz criterion [
Furthermore, according to the biological interpretations of interior equilibrium, phytoplankton population and zooplankton population all exist provided that
In Theorem
Model system (
Firstly, the Jacobian matrix of the model system (
By virtue of (
Based on Routh-Hurwitz criterion [
By denoting the quantities
When the predation rate
By choosing predation rate
Furthermore, in order to show if Hopf bifurcation occurs at
Assuming all the roots of (
Now we will verify the transversality condition:
Substituting
It follows from (
By solving (
If condition (iii) holds, then
By choosing
When
In order to facilitate the following numerical experiments, some preliminaries about Poincaré surface-of-section technique and Lyapunov exponents are introduced in Remarks
A traditional approach to gain preliminary insight into the properties of dynamical system is to carry out a one-dimensional bifurcation analysis. One-dimensional bifurcation diagrams of Poincaré maps present information about the dependence of the dynamics on a certain parameter. The analysis is expected to reveal the type of attractor to which the dynamics will ultimately settle down after passing the initial transient phase and within which the trajectory will then remain forever. On a Poincaré surface of section, the dynamical behavior can be described by a discrete map whose phase-space dimension is less than that of the original continuous flow. Chaotic flows can be understood based on concepts that are convenient for maps such as unstable orbits (see [
A finite number of points corresponds to a periodic solution; that is, one point corresponds to a solution of period equal to that of the forcing term, namely,
A closed curve corresponds to a quasiperiodic solution, that is, a solution consisting of two incommensurate frequencies or, equivalently, having a trajectory that is dense on tours.
A collection of points that is fractal corresponds to chaos, namely, a stranger in phase space; a collection of points that form a cloud that is disorganized, partially organized, or fuzzy may (or may not ) correspond to chaotic attractors.
In a given embedding dimension, the Lyapunov exponent is a measure of the speeds at which initially nearby trajectories of the system diverge. There is a Lyapunov exponent for each dimension of the process, which together constitutes the Lyapunov spectrum for the dynamical system. The Lyapunov exponent is related to the predictability of the system, and the largest Lyapunov exponent of a stable system does not exceed zero. However, a chaotic system has at least one positive Lyapunov exponent. A bounded system with a positive Lyapunov exponent is one operational definition of chaotic behaviors, which presents a quantitative measure of the average rate of separation of nearby trajectories on the attractor. Over the years, a number of methods have been introduced for computation of Lyapunov exponents (see [
In the absence of seasonality, we will perform numerical experiments to observe the dynamics of the model system (
Bifurcation diagram of Poincaré section for the nutrient and predation rate
Bifurcation diagram of Poincaré section for the nutrient and predation rate
According to (
Dynamical response of model system (
Phase portrait of model system (
As
Dynamical response of model system (
A limit cycle corresponding to periodic solution in Figure
Dynamical response of model system (
Phase portrait of model system (
Furthermore, chaotic attractor of model system (
Lyapunove exponents and corresponding Lyapunov dimensions of model system (
| |||||
---|---|---|---|---|---|
0.08 | 0.085 | 0.09 | 0.095 | 0.1 | |
|
0.13761 | 0.1543 | 0.1818 | 0.1913 | 0.2044 |
|
0 | 0 | 0 | 0 | 0 |
|
−1.1265 | −1.5804 | −1.5803 | −1.5893 | −1.5710 |
|
2.1222 | 2.0976 | 2.115 | 2.1204 | 2.1301 |
Chaotic attractor of model system (
Corresponding Poincaré section of model system (
It follows from the Table
Oscillatory population may be driven to extinction in presence of environmental stochasticity when the population density is very low [
Bifurcation diagram of Poincaré section for the nutrient and harvest effort
With the help of MATLAB, numerical experiments with hypothetical set of parameters are performed to understand the dynamical effect of seasonal intrinsic growth rate of nutrient biomass on model system (
Bifurcation diagram of Poincaré section for the nutrient and
It should be noted that
Dynamical response of model system (
Dynamical response of model system (
Corresponding Poincaré section of model system (
It can be observed that the dynamical responses fluctuate with complex structures when the model system admits chaotic behavior. Especially, some dynamical response even comes close to zero within some time interval, which biologically reflects harmful phytoplankton bloom. Consequently, there has been considerable scientific attention towards harmful phytoplankton bloom and its associate control strategy. Based on the theoretical analysis, it is found that high nutrient levels and favorable conditions play a key role in rapid or massive growth of phytoplankton bloom; low nutrient concentration, high predation pressure from zooplankton, and other unfavorable conditions limit phytoplankton growth, which leads to oscillations or recurring bloom in the nutrient-phytoplankton-zooplankton marine ecosystem.
Based on the numerical experiments shown in Figure
It is well known that phytoplankton population and zooplankton population play a key role in large-scale global processes such as ocean-atmosphere dynamics and climate change. Plankton population constitutes the bottom level of the marine and terrestrial life. Recently, harmful algal blooms (HAB) are widely reported and have become a serious environmental problem worldwide as its serious social and economic consequence. Therefore, a better understanding of mechanisms that determine the plankton dynamics is of considerable interest in recent decades [
Compared with the previous related work [
All authors of this paper declare that there is no conflict of interests regarding the publication of this paper. The authors have no proprietary, financial, professional, or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this paper.
The authors gratefully acknowledge anonymous reviewers, editors and Professor Lev Ryashko's comments. This work is supported by National Natural Science Foundation of China, Grant no. 61104003, Grant no. 61273008, and Grant no. 61104093; Research Foundation for Doctoral Program of Higher Education of Education Ministry, Grant no. 20110042120016; Hebei Province Natural Science Foundation, Grant no. F2011501023; Fundamental Research Funds for the Central Universities, Grant no. N120423009; Research Foundation for Science and Technology Pillar Program of Northeastern University at Qinhuangdao, Grant no. XNK201301. This work is supported by State Key Laboratory of Integrated Automation of Process Industry, Northeastern University, supported by Hong Kong Admission Scheme for Mainland Talents and Professionals, Hong Kong, Special Administrative Region.