The Control Data Method: A New Method of Modeling in Population Dynamics

Abstract

A novel modeling method for population dynamics is developed. Based on the classical Lotka-Volterra model, we construct a new predator-prey model with unknown parameters to simulate the behaviors of predator and prey. Using a the approximation property and the machine learning approach of artificial neural networks, a tuning algorithm of unknown parameters is obtained and the factual data of predator-prey can be asymptotically stabilized using a neural network controller. Numerical examples and analysis of the results are presented.

1. Introduction

A mathematical model of population is an abstract model that uses mathematical language to describe the behavior between population and the relationships of populations. Mathematical models are extremely powerful because they usually enable explanation and prediction to be made about population. Soon after Lotka and Volterra’s pioneering works [1, 2], many scholars established a large number of mathematical models to simulate the evolution of population. The theories and applications of mathematical ecology are attended by the most momentous results. Particularly, in the seventies of the last century, the new theories and methods of differential equations and dynamical systems have been widely used in study of population genetics, neurobiology, epidemiology, immunology, physiology, and environmental pollution, and so forth. See the monographs of Murray [3], Chen [4], Freedman [5], and Murray [6] for a detailed description of a population and for various mathematical methods for analyzing population models.

One of the most basic and important models is the Lotka-Volterra type model. In the last decades, considerable work on the permanence, the extinction, and the global asymptotic stability of autonomous or nonautonomous Lotka-Volterra type models have been studied extensively, for example, [7–11] and the references therein. In addition to these, the books of Takeuchi [9], Gopalsamy [12], and Kuang [13] are good sources for the dynamical behavior of Lotka-Volterra models.

Unfortunately, we can never make a completely precise model of a population system, and there are always phenomena which we will not be able to model. Thus, there will always be model errors or model uncertainties. For example, in the establishment of the mathematical model, we ignored a number of secondary factors in order to make an analysis of mathematical reasoning. However, with the rapid progress of science and technology, especially the fast development of computer and network, more information on biological population can be obtained. A large amount of information shows that the existing models are not able to precisely simulate the developmental process of the population. In other words, those secondary factors have been ignored in the modeling process but they played an important role in the development process of populations. Therefore, the traditional models cannot simulate the real dynamic of population and the traditional modeling technique is not perfect.

On the other hand, we note that an artificial neural network (ANN), usually called neural network (NN), is a mathematical model or computational model that is inspired by the structual and/or functional aspects of biological neural networks. In most cases, ANN is an adaptive system that changes its structure based on external or internal information that flows through the network during the learning phase. For this very reason, ANN is found in almost every domain of applied science, such as pattern recognition, data classification, medicine, sales forecasting, industrial process control, customer research, data validation, risk management, and target marketing.

ANN based on biological neural systems has been studied for years, and their properties of learning and adaptation, classification, function approximation, feature extraction, and more have made them of extreme use in signal processing and system identification applications. These are open-loop applications, and the theory of ANN has been very well-developed in this area. The applications of ANN in closed-loop feedback control systems have only recently been rigorously studied, and a foundation for neural networks in control has been provided in seminal results by Narendra et al. [14–17] and others. Several researchers have studied ANN control and managed to prove stability [18–20]. In particularly, Jiang [21] presented a neural network control scheme for tracking a nonlinear system using trajectory tracking, and an iterative training law described by a positive definite discrete kernel is also presented. Hayakawa [22] considered a neural network hybrid adaptive control framework for nonlinear uncertain hybrid dynamical systems.

Motivated by these considerations, in this paper, we will propose a new modeling method, using machine learning of the ANN theory to amend Lotka-Volterra predator-prey model such that the revised model can better simulate or control the development of populations.

This paper is organized as follows. Using machine learning of the ANN theory, a new predator-prey model is introduced in Section 2. In Section 3, we give some functional-link neural network (FLNN) weight tuning algorithms for this model. In Section 4, some specific examples are given to illustrate our results. Finally, Section 5 presents the conclusions.

2. Model Description and Preliminaries

It is well known that Lotka-Volterra models are fundamental population models. For example, the following classical autonomous model is used to model the interaction of the prey and predator: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑏₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂𝑑𝑥₂𝑑𝑡=𝑥₂−𝑏₂+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂, (1) where 𝑥₁ and 𝑥₂ represent the population densities of prey and predator at time 𝑡, respectively; 𝑏_𝑖 and 𝑎_𝑖𝑖 (𝑖=1,2) are nonnegative constants. The dynamic behaviors of model (1) are clear. It has no periodic solution in region ℝ²₊={(𝑥₁,𝑥₂)∶𝑥₁≥0,𝑥₂≥0} and has a saddle (0,0) and a stable nod (𝑏₁/𝑎₁₁,0) under the condition 𝑏₁𝑎₂₁≤𝑏₂𝑎₁₁, whereas it has two saddles (0,0) and (𝑏₁/𝑎₁₁,0) and one stable focus ((𝑏₁𝑎₂₂+𝑏₂𝑎₁₂)/(𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁),(𝑏₁𝑎₂₁−𝑏₂𝑎₁₁)/(𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁)) under the condition 𝑏₁𝑎₂₁>𝑏₂𝑎₁₁ as shown in Figure 1.

(a)

(b)

Figure 1 

The dynamical behavior of model (1): (a) a stable focus ((𝑟1𝑎22+𝑟2𝑎12)/(𝑎11𝑎22+𝑎12𝑎21),(𝑟1𝑎21−𝑟2𝑎11)/(𝑎11𝑎22+𝑎12𝑎21)); (b) a stable nod (𝑟1/𝑎11,0).

Along with the development of technology, more information on biological population can be obtained. We find that the model (1) could not simulate the real dynamic behavior of predator and prey. Therefore, according to the traditional Lotka-Volterra predator-prey model, we construct a new predator-prey model to simulate the states of predator and prey. The model takes the following form: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑏₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂+𝜏₁𝑑𝑥₂𝑑𝑡=𝑥₂−𝑏₂+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂+𝜏₂, (2) where 𝜏=(𝜏₁,𝜏₂)^𝑇 is an unknown function.

Next, we will use the machine learning of the ANN theory to approximate the unknown function 𝜏, such that model (2) can better simulate the behaviors of predator and prey.

Let 𝑥=(𝑥₁,𝑥₂) be a solution of model (2). Suppose that the factual data of predator-prey is 𝑥_𝑑=(𝑥_1𝑑,𝑥_2𝑑)∈𝑅²₊. Then, we define the error function 𝑒(𝑡) by 𝑒(𝑡)=𝑥−𝑥_𝑑=𝑥₁−𝑥_1𝑑,𝑥₂−𝑥_2𝑑^𝑇. (3) Since 𝑥_𝑑 is the factual data, so 𝑥_𝑑 satisfies the following assumption.

(𝐴₁) The factual data is bounded, that is, 𝑥_𝑑,𝑑𝑥_𝑑𝑑𝑡≤𝑋_𝐵, (4) where 𝑋_𝐵 is a known positive constant and 𝑑𝑥_𝑑=𝑑𝑡𝑑𝑥_1𝑑,𝑑𝑡𝑑𝑥_2𝑑𝑑𝑡^𝑇. (5)

By differentiating (3) and invoking (2), it is seen that the model is expressed in the terms of a tracking error as 𝑑𝑒𝑦𝑑𝑡=−𝐷𝑒+𝑝+𝜏, (6) where 𝑑𝑒=𝑑𝑡𝑑𝑒₁,𝑑𝑡𝑑𝑒₂𝑑𝑡^𝑇, (7)𝑦=(𝑥_1𝑑,𝑥_2𝑑,𝑒₁,𝑒₂)^𝑇, 𝐷=diag(0,−𝑏₂) is diagonal matrix, and𝑝𝑦=𝑝₁𝑦𝑝₂𝑦=𝑥_1𝑑+𝑒₁𝑏₁−𝑎₁₁𝑥_1𝑑+𝑒₁−𝑎₁₂𝑥_2𝑑+𝑒₂−𝑥̇_1𝑑𝑥_2𝑑+𝑒₂𝑎₂₁𝑥_1𝑑+𝑒₁−𝑎₂₂𝑥_2𝑑+𝑒₂−𝑏₂𝑥_2𝑑−𝑥̇_2𝑑. (8)Let 𝑆 be a compact, simply connected set of 𝑅⁴ and 𝑝(⋅)∶𝑆→𝑅². Define 𝐶²(𝑆) as a space of continuous functions 𝑝(⋅). The universal approximation theorem claims that, for real valued function 𝑝(𝑦), there exists an ideal approximating weight 𝑊 such that𝑝𝑦=𝑊^𝑇𝜙𝑦+𝜀 (9) with the estimation tracking error bounded by ‖𝜀‖<𝜀_𝑌, (10) and 𝜙(⋅)∶𝑅⁴→𝑅² is a basis set which can choose 1(),11+exp−⋅1+exp(−⋅)^𝑇 (11) or (𝑥₁−𝑥_1𝑑,𝑥₂−𝑥_2𝑑)^𝑇. One must notice as well that the ideal weight is unknown and not even unique. Assume that ideal weight is a constant and bounded so that ‖𝑊‖_𝐹≤𝑊_𝐵 (12) with the bounded 𝑊_𝐵 known, where ‖𝑊‖_𝐹=tr(𝐴^𝑇𝐴) is the Frobenius norm of 𝑊 and tr(⋅) is the matrix trace, that is, sum of diagonal elements. For more details, we refer to [23, 24].

Then, an estimate of function 𝑝(𝑦) is given by 𝑝𝑦=𝑊^𝑇𝜙𝑦 (13) with 𝑊 as the current actual values of the one-layer functional-link neural network (FLNN) controller weights as provided by the tuning algorithm to be specified. It is necessary to show how to tune the ANN weight 𝑊 on-line so as to guarantee stable tracking. The tuning algorithm found will presumably modify the actual weight 𝑊 so that they become close to the ideal weight 𝑊, which is unknown. We choose a general sort of approximation-based function, which is derived by setting 𝜏(𝑡)=−𝑊^𝑇𝜙𝑦−𝐾_𝑣𝑒, (14) where 𝐾_𝑣𝑒 is an outer proportional-plus-derivative tracking loop. Using this controller, system (6) is rewritten by the following form: 𝑑𝑒𝑑𝑡=−𝐷𝑒−𝐾_𝑣𝑒+𝑊^𝑇𝜙𝑦+𝜀, (15) where 𝑊=𝑊−𝑊 is the function approximation error.

3. FLNN Weight Tuning Algorithms for Model (2)

In this section, we give some FLNN weight tuning algorithms that guarantee the tracking stability of model (15). It is required to demonstrate that the tracking error 𝑒(𝑡) is suitably small and that the FLNN weights 𝑊 remain bounded, for that, the unknown function 𝜏(𝑡) is bounded.

3.1. Ideal Case

In this subsection, we detail the behavior in the idealized case, where net functional reconstruction error 𝜀 is zero. The following theorem derives the ANN control 𝜏(𝑡) that asymptotically stabilized the predator-prey model about its desired trajectory 𝑥_𝑑(𝑡).

Theorem 1. Suppose that assumption (𝐴₁) holds and 𝜀_𝑌 in (10) is equal to zero. Suppose that the unknown function 𝑝(𝑦) be given by (9) and ANN weight tuning provided by 𝑑𝑊𝑦𝑒𝑑𝑡=𝐹𝜙^𝑇, (16) where 𝐹=𝐹^𝑇>0 is a constant design parameter matrix. Then the tracking error 𝑒(𝑡) goes to zero with 𝑡 and the weight estimates 𝑊 are bounded.

Proof. Under the ideal case, the error system is 𝑑𝑒𝑑𝑡=−𝐷𝑒−𝐾_𝑣𝑒+𝑊^𝑇𝜙𝑦. (17) Select the Liapunov function candidate 𝑉(𝑡)=12𝑒^𝑇1𝑒+2tr𝑊^𝑇𝐹⁻¹𝑊. (18) Calculating the derivative of 𝑉(𝑡) along the error system (17), it follows that 𝑑𝑉𝑑𝑡=𝑒^𝑇𝑑𝑒𝑑𝑡+tr𝑊𝐹⁻¹𝑑𝑊𝑑𝑡=−𝑒^𝑇𝐷+𝐾_𝑣𝑒+tr𝑊𝐹⁻¹𝑑𝑊𝑦𝑒𝑑𝑡+𝜙^𝑇. (19) Since 𝑊=𝑊−𝑊 and 𝑊 is constant, from (16), it yields that 𝑑𝑊𝑦𝑒𝑑𝑡=−𝐹𝜙^𝑇. (20) From this and (19), it follows that 𝑑𝑉𝑑𝑡=−𝑒^𝑇𝐷+𝐾_𝑣𝑒. (21) Since 𝑉(𝑡)>0 and 𝑑𝑉/𝑑𝑡≤0, this shows stability in the sense of Liapunov so that 𝑒(𝑡) and 𝑊 (and hence 𝑊) are bounded. Thus −^∞₀𝑑𝑉<∞. (22) Now 𝑑²𝑉𝑑𝑡²=−2𝑒^𝑇𝐷+𝐾_𝑣𝑑𝑒𝑑𝑡 (23) and the right-hand side of (16) verifies the boundedness of 𝑑𝑒/𝑑𝑡 and hence of 𝑑²𝑉/𝑑𝑡² and therefore the uniform continuity of 𝑑𝑉/𝑑𝑡. This allows us to invoke Barbalat’s Lemma in connection with (23) to conclude that 𝑑𝑉/𝑑𝑡 goes to zero with 𝑡 and hence that 𝑒(𝑡) vanishes as 𝑡→∞. The proof is complete.

On model (2), we have Theorem 2 that is a direct consequence of Theorem 1.

Theorem 2. Suppose that assumption (𝐴₁) holds and 𝜀_𝑌 in (10) is equal to zero. Further, let 𝜏(𝑡)=−𝑊^𝑇𝜙(𝑦)−𝐾_𝑣𝑒, where 𝑊 satisfy condition (16). Then, the error 𝑒(𝑡)=𝑥−𝑥_𝑑→0, with 𝑡→∞.

Remark 3. From Theorem 2, we note that the solution of model (2) goes to the factual data 𝑥_𝑑. So, model (2) can better simulate the dynamical behaviors of predator and prey.

3.2. The Nonideal Case

It has just been seen that there is no ANN functional approximation error under the ideal case. In this subsection, it will be seen that if the ANN approximation errors are not zero but bounded, then the tracking errors do not vanish but are bounded by small enough values to guarantee good tracking performance.

Theorem 4. Suppose that assumption (𝐴₁) holds and 𝜀_𝑌 in (10) is constant. Let the unknown function 𝑝(𝑦) be given by (9) and let NN weight tuning be provided by 𝑑𝑊𝑦𝑒𝑑𝑡=𝐹𝜙^𝑇, (24) where 𝐹=𝐹^𝑇>0 is a constant design parameter matrix. Then the tracking error 𝑒(𝑡) is uniform ultimate boundedness and the weight estimates 𝑊 are bounded. Moreover, 𝑒(𝑡) may be kept as small as desired by increasing the gain 𝐾_𝑣.

Proof. Let the NN approximation property (9) hold for the function 𝑝(𝑦) given in (8) with a given accuracy 𝜀_𝑌 for all 𝑦 in the compact set 𝑆_𝑦={𝑦∶‖𝑦‖<𝑏_𝑦} with 𝑏_𝑦>𝑋_𝐵.
Select the Liapunov function candidate 𝑉(𝑡)=12𝑒^𝑇1𝑒+2tr𝑊^𝑇𝐹⁻¹𝑊. (25) Calculating the derivative of 𝑉(𝑒) along the error system (15), it follows that 𝑑𝑉𝑑𝑡=𝑒^𝑇𝑑𝑒𝑑𝑡+tr𝑊𝐹⁻¹𝑑𝑊𝑑𝑡=−𝑒^𝑇𝐷+𝐾_𝑣𝑒+tr𝑊𝐹⁻¹𝑑𝑊𝑦𝑒𝑑𝑡+𝜙^𝑇+𝑒^𝑇𝜀. (26) Since 𝑊=𝑊−𝑊 and 𝑊 is constant, from (24) it yields that 𝑑𝑊𝑦𝑒𝑑𝑡=−𝐹𝜙^𝑇. (27) From this and (26), it follows that 𝑑𝑉𝑑𝑡≤−𝜆_min‖𝑒‖²+𝜀_𝑌‖𝑒‖, (28) where 𝜆_min is the minimum singular value of matrix (𝐷+𝐾_𝑣). Since 𝜀_𝑌 is constant, 𝑑𝑉/𝑑𝑡≤0 as long as ‖𝑒‖>𝜀_𝑌𝜆_min∶=𝐿. (29)
Let 𝑆_𝑎={𝑒∶𝑒∈𝑆_𝑦,𝑉(𝑒)≤𝐶_𝑎}, where 𝐶_𝑎 is the maximum positive constant such that 𝑆_𝑎∈𝑆_𝑦; 𝑆_𝑏={𝑒∶𝑒∈𝑆_𝑦,𝑉(𝑒)≤𝐶_𝑏,𝑑𝑉/𝑑𝑡>0}, where 𝐶_𝑏 is the maximum positive constant such that 𝑆_𝑏∈𝑆_𝑦. It is obviouse that 𝑆_𝑎⊂𝑆_𝑏.
Suppose that initial tracking error 𝑒(0)∈𝑆_𝑎. If 𝑒(0)∈𝑆_𝑎⧵𝑆_𝑏, 𝑑𝑉/𝑑𝑡<0 by the definitions of 𝑆_𝑎 and 𝑆_𝑏. Therefore, 𝑉 will gradually become smaller until the access to 𝑆_𝑏. At this time, 𝑉 will become bigger from the definition 𝑆_𝑏. So, it follows that 𝑒(𝑡) is the asymptotic convergence to the border of 𝑆_𝑏. Similarly, if 𝑒(0)∈𝑆_𝑏, we can obtain the same results.
Now, let 𝑆={𝑒∶‖𝑒‖<𝐿}. Then we choose appropriate 𝐾_𝑣 and 𝜀^∗<𝜀_𝑌 such that 𝐿<𝜌 for all 𝜀<𝜀^∗, where 𝜌 is the diameter of set 𝑆_𝑦. From this, we have 𝑇⊂𝑆_𝑦 and 𝑆_𝑏⊂𝑇. Otherwise, there is a point 𝑒^∗ in set 𝑆_𝑦 such that 𝑒^∗∈𝑆. Hence, we have 𝑉̇(𝑒^∗)<0. This is a conflict with the definition of set 𝑆.
From the above discussion, we get that if 𝑒(0)∈𝑆_𝑎, then 𝑒(𝑡) is the asymptotic convergence to the border of 𝑆_𝑏. On the other hand, since 𝑆_𝑏∈𝑆, it follows that ‖𝑒‖<𝐿 as 𝑡→∞. Therefore, tracking error 𝑒(𝑡)  is uniform ultimate boundedness and the boundedness 𝐿 may be kept as small as desired by increasing the gain 𝐾_𝑣. This completes the proof.

From Theorem 4, on the model (2), we have the following Theorem 5.

Theorem 5. Suppose that assumption (𝐴₁) holds and 𝜀_𝑌 in (10) is constant. Further, let 𝜏(𝑡)=−𝑊^𝑇𝜙(𝑦)−𝐾_𝑣𝑒, where 𝑊 satisfies condition (24). Then, the tracking error 𝑒(𝑡)=𝑥−𝑥_𝑑 is uniform ultimate boundedness and the wight estimates 𝑊 are bounded. Moreover, 𝑒(𝑡) may be kept as small as desired by increasing the gain 𝐾_𝑣.

Remark 6. From Theorem 5, we can get that the solution of model (2) goes to the factual data 𝑥_𝑑. So, if the conditions of Theorem 5 hold, then model (2) can better simulate the behaviors of predator and prey.

4. Example and Numerical Simulation

In this paper, we proposed a new modeling method, using machine learning of the ANN theory to amend the traditional Lotka-Volterra model such that the revised model can better simulate or control the behaviors of population.

In order to testify the validity of our results, we consider the following model: 𝑑𝑥₁𝑑𝑡=𝑥₁𝑏₁−0.5𝑥₁−2𝑥₂+𝜏₁,𝑑𝑥₂𝑑𝑡=𝑥₂−2+2𝑥₁−0.2𝑥₂+𝜏₂, (30) where 𝜏=(𝜏₁,𝜏₂) is an unknown function.

If we choose 𝑏₁=1.5 and 𝜏=0 in model (30), it is clear that model (30) has a saddle (0,0) and a stable nod (𝑟₁/𝑎₁₁,0)=(3,0) which are shown in Figures 2(a) and 3(a).

(a)

(b)

Figure 2 

The trajectory of the prey 𝑥1 of model (30) with (a) 𝑏1=1.5 and 𝜏=0; (b) 𝑏1=1.5, 𝜏≠0, and 𝑥𝑑=(1.5,0.5).

(a)

(b)

Figure 3 

The trajectory of the predator 𝑥2 of model (30) with (a) 𝑏1=1.5 and 𝜏=0; (b) 𝑏1=1.5, 𝜏≠0, and 𝑥𝑑=(1.5,0.5).

However, if a factual data of predator-prey is 𝑥_𝑑(𝑡)=(𝑥_1𝑑,𝑥_2𝑑)=(1.5,0.5), then model (30) with 𝑏₁=1.5 cannot simulate the real dynamic of populations. So, in this way, according to the Theorems 1 and 4, we can choose 𝐹=0.30.10.10.3,𝐾_𝑣=50303050, (31) and 𝜙(𝑡)=(𝑥₁−𝑥_1𝑑,𝑥₂−𝑥_2𝑑)^𝑇. Then, we note that the factual data of predator-prey 𝑥_𝑑(𝑡) is asymptotically stable that is, the tracking error 𝑒(𝑡) goes to zero as 𝑡→∞. That is, if we choose 𝜏(𝑡)=−𝑊^𝑇𝜙(𝑦)−𝐾_𝑣𝑒, then model (30) can better simulate (or control) the real behaviors of population which are shown in Figures 2(b) and 3(b).

Further, if we choose 𝑏₁=3.0 and 𝜏=0 in model (30), it is easy to demonstrate that model (30) has two saddles (0,0) and (6,0) and one stable focus (4.18,0.45) which are shown in Figures 4(a) and 5(a).

(a)

(b)

Figure 4 

The trajectory of the prey 𝑥1 of model (30) with (a) 𝑏1=3.0 and 𝜏=0; (b) 𝑏1=3.0, 𝜏≠0, and 𝑥𝑑=(2.5+0.2sin(𝑡),2.0+0.2cos(𝑡)).

(a)

(b)

Figure 5 

The trajectory of the prey 𝑥2 of model (30) with (a) 𝑏1=3.0 and 𝜏=0; (b) 𝑏1=3.0, 𝜏≠0, and 𝑥𝑑=(2.5+0.2sin(𝑡),2.0+0.2cos(𝑡)).

If a factual data of predator-prey is 𝑥_𝑑(𝑡)=𝑥_1𝑑𝑥_2𝑑=(𝑡)(𝑡)2.5+0.2sin2.0+0.2cos, (32) then model (30) with 𝑏₁=3.0 cannot simulate the real dynamic of populations. So, in this way, according to Theorems 1 and 4, we choose 𝐹=1.50.60.62.0,𝐾_𝑣1=300510360, (33) and 𝜙(𝑡)=(𝑥₁−𝑥_1𝑑,𝑥₂−𝑥_2𝑑)^𝑇. Then, we note that the factual data of predator prey, 𝑥_𝑑(𝑡), is asymptotically stable and has asymptotic phase property. That is, if we choose 𝜏(𝑡)=−𝑊^𝑇𝜙(𝑦)−𝐾_𝑣𝑒, then model (30) can better simulate (or control) the real behaviors of predator and prey, which are shown in Figures 4(b) and 5(b). Moreover, the tracking error 𝑒(𝑡) may be kept as small as desired by increasing the gain 𝐾_𝑣 which is shown in Figures 6(a) and 6(b).

(a)

(b)

Figure 6 

The trajectory of error system 𝑒(𝑡): (a) 𝑒1(𝑡) with 𝐾𝑣1 and 𝐾𝑣2, (b) 𝑒2(𝑡) with 𝐾𝑣1 and 𝐾𝑣2; where 𝐾𝑣1=300510360 and 𝐾𝑣2=500510560.

5. Conclusion

According to more information on biological population that can be obtained, we find that the traditional models cannot simulate the real dynamic of populations. Therefore, the traditional modeling technique is not perfect. So, in this paper, based on the traditional and most important Lotka-Volterra model, we developed a new modeling method, using machine learning of the ANN theory to construct a new predator-prey model to simulate the states of predator and prey. From the neural networks function approximation property and a factual data of predator-prey, we proposed a neural network trajectory tracking strategy, and the tuning algorithm of the new model is obtained. That is, under general assumptions, we proved that the tracking error is ultimately uniformly bounded and that the corresponding ultimate bound can be sufficiently decreased by modifying the feedback gain matrix. So, the new model can better simulate or control the real behaviors of populations. Finally, numerical examples were presented to show that the proposed method is feasible and efficient.

Acknowledgments

This research has been partially supported by the Natural Science Foundation of Xinjiang (Grant no. 2011211B08), the Scientific Research Programs of Colleges in Xinjiang (Grant no. XJEDU2011S08), the National Natural Science Foundation of China (Grants nos. 11001235, 11271312, and 11261056), the China Postdoctoral Science Foundation (Grants nos. 20110491750 and 2012T50836).