Optimal Breeding Strategy for Livestock with a Dynamic Price

Abstract

China’s livestock output has been growing, but domestic livestock products such as beef, mutton and pork have been unable to meet domestic consumers’ demands. The imbalance between supply and demand causes unstable livestock prices and affects profits on livestock. Therefore, the purpose of this paper is to provide the optimal breeding strategy for livestock farmers to maximize profits and adjust the balance between supply and demand. Firstly, when the price changes, livestock farmers will respond in two ways: by not adjusting the scale of livestock with the price or adjusting the scale with the price. Therefore, combining the model of price and the behavior of livestock farmers, two livestock breeding models were established. Secondly, we proposed four optimal breeding strategies based on the previously studied models and the main research method is Pontryagin’s Maximum Principle. Optimal breeding strategies are achieved by controlling the growth and output of livestock. Further, their existence was verified. Finally, we simulated two situations and found the most suitable strategy for both situations by comparing profits of four strategies. From that, we obtained several conclusions: The optimal strategy under constant prices is not always reasonable. The effect of price on livestock can promote a faster balance. To get more profits, the livestock farmers should adjust the farm’s productivity reasonably. It is necessary to calculate the optimal strategy results under different behaviors.

1. Introduction

As the largest livestock breeding country in the world, China’s livestock output has been growing. However, with the growth in living standards, the per capita meat consumption in China continues to increase [1]. The per capita beef consumption in China increased from 1.5 kg in 2014 to 2.3 kg in 2020, and the per capita mutton consumption also increased from 1.0 kg in 2014 to 1.2 kg in 2020 [2], so domestic livestock products such as beef and mutton have been unable to meet domestic consumers’ demands [3]. For example, China is not only the largest mutton producer, but the largest mutton consumer [4]. Mutton has great potential in the consumption market, and mutton consumption in China will continue to increase in the future [5,6], but as shown in Figure 1 the domestic supply of mutton is insufficient. The main problem of China’s mutton market is how to promote market balance [7]. The imbalance between supply and demand causes unstable livestock prices and affects the profits of livestock farmers. When the price increases, some farmers will expand their scale of livestock to gain more profits. However, if they increase their scale blindly, it may lead to losses rather than profits [8]. Obviously, this is due to the inappropriate strategies of farmers. Therefore, our goal was to find the best breeding strategy for farmers and achieve a balance between supply and demand for market stability. The strategy research of breeding mode is a very important topic. With the spread of major animal diseases, some scholars have studied the dynamic model of animal diseases [9,10], but its internal mechanism is more complicated, so comprehensive research by means of dynamics and economics is needed.

On the research of optimal breeding scale, Abakumov studied the single-species harvesting model in 1993, established objective functions with no cost and cost to find the optimal scale and got the result of optimal harvesting strategies [11]. Khaltar extended the cost objective function and the zero-cost optimization model of livestock, added the employment cost and studied the labor recruitment model of cattle farm [12,13]. Besides, he determined the structure of optimal impulsive control for optimal processes with impulse control. These articles got the optimal sale rate and breeding scale by using Pontryagin’s maximum principle, but they all ignored the change of price and the effect of price on livestock scale. Pontryagin’s maximum principle is widely used in various practical problems, for example, for the outbreak of the novel coronavirus (COVID-19) that swept across the world [14,15]. Pontryagin’s maximum principle was used to study how to control the spread of COVID-19 [16]. Therefore, in this paper, combined with the dynamic price, we also used Pontryagin’s maximum principle to solve the problem.

When studying single-species models, many domestic and foreign scholars have found that prices cannot be used as the constants in practice and have worked out several forms of prices. Clark and Chen regarded price as a dynamic change of supply; they put forward price algebraic models [17,18]. Then, numerous articles [19,20,21,22] studied those models. However, the price in algebraic form did not conform to the market rules, and market price increases when there is more demand than offer, according to classical economic theory, so according to the effect between demand and supply on prices, Shone put forward a differential price model $\frac{dp}{dt}=\beta (q^d-q^s)$ where $q^d$ is the demand function, $q^s$ is the supply function and $\beta$ is a positive parameter relative to the speed of varying price [23]. Combining linear demand functions, nonlinear demand functions and actual supply functions, the authors of [24,25,26,27,28,29,30] and others have studied various forms of price differential models and conducted a series of dynamics studies and practical applications. We refer to the price model $\frac{dp}{dt}=\beta (A-p-qnE)$ in [25], and then consider the interaction between price and breeding quantity in combination with farmers’ behavior, which has not been considered in previous studies.

In this paper, thinking about the interaction between price and breeding quantity, we use mathematical methods to study how to formulate optimal breeding strategies for livestock farmers to achieve maximum profits and balance between supply and demand. Mathematical methods have the potential to analyze the mechanisms of livestock growth and the changing of prices, and can put forward quantitative optimal breeding strategies. The organization of this paper is as follows. In Section 2, we set up a price model regulated by supply and demand and two livestock models according to the behavior of livestock farmers. In Section 3, we propose four optimal breeding strategies based on the previously studied models by applying Pontryagin’s maximum principle. The existence of strategies are proved, and two specific examples are given in Section 4. Finally, in Section 5, there are some conclusions about breeding strategies.

2. Mathematical Model

In this section, the model without price adjustment and the model with price adjustment are put forward. First, we give the most basic livestock model. Denote $N(t)$ to be the total number of livestock animals on farms at time t. Since the total number of livestock animals is limited by many factors—breeding environment, breeding feed, breeding technology, etc.—a logistic model should be considered for modeling N.

$$\frac{dN}{dt}=rN(1-\frac{N}{K})-g(N),$$

(1)

where r is the intrinsic growth rate, K is the maximum breeding amount of livestock, $g(N)$ is the supply function and $g(N)$ indicates the number of livestock animals put into the market at time t.

According to economics, when supply and demand are in balance, the price is constant, but when supply and demand are not in balance, the price regulated by supply and demand is changing. It is assumed that the demand function is $f(p)$. Based on the facts that mutton is non-essential and demand decreases when the price rises, $f(p)$ is a monotonically decreasing function of price and satisfies $\underset{p\to 0}{lim}f(p)=D$, where D is the maximum demand. In our model, $f(p)$ takes the form of $D-bep$, where b is the price adjustment coefficient, e is the average weight of livestock when it is put on the market and $\varphi$ is the proportional coefficient (see for reference, [30]). Combining the supply function $g(N)$, the price model is

$$\frac{dp}{dt}=\varphi (D-bep-g(N)),$$

(2)

In the face of dynamic prices, livestock farmers may not regulate their breeding scale with price, or they may regulate their breeding scale with price by adjusting their own pastures’ productivity or buying and selling young livestock. Therefore, we need to establish the different livestock models based on different behaviors.

2.1. The Model without Price Adjustment

When the price is changing, livestock farmers may not regulate their breeding scale with price, which means that N is not affected by the price p. Therefore, the model is

$$\left\{\begin{array}{c}\frac{dN}{dt}=rN(1-\frac{N}{K})-\alpha N,\hfill \\ \frac{dp}{dt}=\varphi (D-bep-\alpha N),\hfill \\ N(0)\ge 0,p(0)\ge 0,0\le \alpha <r,\hfill \end{array}\right.$$

(3)

where $\alpha$ is the sale rate of livestock on farms to the market and $\alpha N$ is the total number of livestock animals put on the market from the farms. In other words, $\alpha N$ is also the supply, so $g(N)=\alpha N$. The analysis of the system (3) is as follows. The positively invariant set, the existence of positive equilibrium and stability of equilibrium are given.

Theorem 1.

For $\forall t>0$, the solution of the system is positive and bounded, if the initial conditions of the system are $N(0)$ and $p(0)$ being non-negative. The positively invariant set and attraction domain of the system are

$$\Gamma =\left\{\left(N,p\right)\in R_{+}^2:N\le K,p\le \frac{D}{be}\right\}.$$

Theorem 2.

There are two equilibriums in the system (3): $A=(N_1,p_1)=(0,\frac{D}{be})$, $B=(N_2,p_2)=((r-\alpha )\frac{K}{r},\frac{D-(r-\alpha )\frac{\alpha K}{r}}{be})$. The equilibrium B is positive if each parameter of the system satisfies (a) or (b), where

(a) 

$0\le \alpha <r,$ if $Kr-4D\le 0$,

(b) 

$0\le \alpha <\frac{rK-\sqrt{Kr(Kr-4D)}}{2K}$ or $\frac{rK+\sqrt{Kr(Kr-4D)}}{2K}<\alpha <r,$ if $Kr-4D>0$.

Theorem 3.

The equilibrium A of the model (3) is unstable, and the equilibrium B is globally asymptotically stable.

The proofs of the above theorems are given in Appendix A. Since the equilibrium A has no practical significance, our study only focuses on the equilibrium B in the following.

2.2. The Model with Price Adjustment

When the price changes, livestock farmers will respond in two ways. If livestock farmers regulate their breeding scale by adjusting their own farms’ productivity or buying and selling young livestock, there will be a direct impact on the breeding scale, and this direct effect will be caused by the subjective perceptions of livestock farmers. If the price increases, livestock farmers will expand the breeding scale in order to gain more profits, and if the price decreases, livestock farmers will reduce the breeding scale in order to reduce losses [8]. This shows that the adjustment of price is proportional to N, so we express this impact in the form of $\epsilon e(p-p_0)$, where $\epsilon$ is adjustment coefficient of livestock impacted by price and $p_0$ is the balanced price determined by farms. In Section 3, we calculate the optimal value of $\epsilon$ in order to provide livestock farmers with the optimal breeding strategy. The model is

$$\left\{\begin{array}{c}\frac{dN}{dt}=rN(1-\frac{N}{K})+\epsilon e(p-p_0)-\alpha N,\hfill \\ \frac{dp}{dt}=\varphi (D-bep-\alpha N),\hfill \\ N(0)\ge 0,p(0)\ge 0.\hfill \end{array}\right.$$

(4)

All parameters of system (4) have the same meanings as those of system (3). In the following, the positively invariant set, the existence and stability of positive equilibrium are given.

Theorem 4.

The solutions of model (4) with non-negative initial conditions remain non-negative and bounded for all time $t>0$. The positively invariant set and attraction domain of the system are

$$\Gamma =\left\{\left(N,p\right)\in R_{+}^2:N\le \frac{K}{2}+\frac{\sqrt{K^2{r}^2{b}^2+4r\epsilon DKb}}{2rb},p\le \frac{D}{be}\right\}.$$

Theorem 5.

There is a positive equilibrium $(N^{*},p^{*})$ of the system (4), if each parameter of the system satisfies

$$2Drb-\alpha K(rb-\epsilon \alpha -b\alpha +\sqrt{{(rb-\epsilon \alpha -b\alpha )}^2+4\frac{rb\epsilon }{K}(D-be{p}_0)})>0.$$

Theorem 6.

The equilibrium $(N^{*},p^{*})$ of the model (4) is locally asymptotically stable.

The proofs of the above theorems are given in Appendix B.

3. Optimal Breeding Strategy

In this section, four optimal breeding strategies under each mode are obtained. One of the breeding strategies is to consider how the livestock scale should be adjusted when the price is constant. When the price changes, the other three strategies are formulated according to the three behaviors of livestock farmers: not adjusting the scale of livestock with the price, adjusting livestock productivity and buying or selling young livestock with the price. Based on previously studied models, and the specific income and expenditure situation, the profit function under each mode is established. The generalized profit function is

$$J(\alpha ,p,N)=\underset{0}{\overset{T}{\int }}(ep\alpha (t)N(t)-CN(t)-M(t))dt.$$

(5)

According to cost–benefit theory, the profit function is the difference between the benefit and the cost. In breeding, we assume that the main revenue is from the sale of livestock, e is the average weight of livestock put on the market, and the price of an animal is $ep$, so the income is $ep\alpha N$. The cost of breeding includes the costs of materials, services, labor and land. We suppose these costs are averaged as C (each animal per unit time). $M(t)$ represents other expenses or income. T is the set terminal time, which suggests that our study considers the optimal breeding strategy in the time period 0 to T.

Our goal is to maximum the profit function (5) and achieve the optimal scale of breeding by regulating the growth and the sale rate of livestock. When livestock farmers adopt different behaviors, the optimal sale rate $\alpha$, the optimal adjustment rate $\epsilon$, the final balance price p and the optimal breeding scale N are obtained by using Pontryagin’s maximum principle. Under the optimal breeding strategy, the supply–demand equilibrium is also reached.

3.1. Optimal Breeding Strategy with Constant Price

Assuming that the price of livestock is constant without being regulated by supply and demand, the optimal breeding strategy is given. Without other expenses or income, so $M(t)=0$, the payment function in this time period is:

$$J(\alpha ,N)=\underset{0}{\overset{T}{\int }}(ep\alpha (t)N(t)-CN(t))dt,$$

(6)

subject to

$$\frac{dN}{dt}=G(N)-\alpha (t)N=rN(1-\frac{N}{K})-\alpha (t)N,$$

(7)

$$N(0)\ge 0,0\le \alpha (t)\le r.$$

Our goal is to find ${\alpha }^{*},N^{*}$ to maximize the above profit function, namely,

$$J({\alpha }^{*},N^{*})=max\{J(\alpha ,N)|\alpha \in U\},$$

(8)

where

$$U=\{\alpha |0<\alpha <r\}.$$

U is the control set.

Theorem 7.

The optimal control formula is

$${\alpha }^{*}=\frac{1}{2}(r+\frac{C}{p}),N^{*}=(r-\frac{C}{ep})\frac{K}{2r}.$$

If ${\alpha }^{*}\in U$, the optimal strategy is

$$\alpha =\left\{\begin{array}{c}0\hfill \\ {\alpha }^{*}\hfill \\ r\hfill \end{array}\right.\begin{array}{c}N(0)<N^{*},\\ N(0)=N^{*},\\ N(0)>N^{*}.\end{array}$$

If ${\alpha }^{*}>{\alpha }_{max}=r$, the optimal strategy is to use the maximum sale rate. If ${\alpha }^{*}<0$, the optimal strategy is not to sell livestock.

Proof. 

The Pontryagin maximization principle [31] is used to solve the necessary conditions for optimal breeding results. According to the principle of maximization, solving the optimal control problem is equivalent to solving the problem of maximizing Hamiltonian. The Hamiltonian is constructed as

$$H=(ep\alpha N-CN)+{\lambda }_1\left[rN(1-\frac{N}{K})-\alpha N\right]=(ep-{\lambda }_1)\alpha N-CN+{\lambda }_{1}rN(1-\frac{N}{K}),$$

(9)

where ${\lambda }_1$ is an adjoint variable, which satisfies

$$\frac{\partial {\lambda }_1}{\partial t}=-\frac{\partial H}{\partial N}=-ep\alpha +{\lambda }_1\alpha +C-{\lambda }_1{G}^{\prime }(N),$$

$${\lambda }_1(T)=0.$$

Obviously, H is linear with respect to the control variable $\alpha$, and the control is of bang–bang type [32], so the coefficient of H with respect to $\alpha$ is defined as

$$\sigma (N,t)=(ep-{\lambda }_1)N.$$

To maximize H, then,

$$\alpha =\left\{\begin{array}{c}0\hfill \\ uncertain\hfill \\ r\hfill \end{array}\right.\begin{array}{c}\sigma (N,t)<0,\\ \sigma (N,t)=0,\\ \sigma (N,t)>0.\end{array}$$

The optimal control solution under uncertainty is discussed as follows. If $\sigma (N,t)=0$,

$$\frac{\partial H}{\partial \alpha }=\sigma (N,t)=0,$$

$$ep={\lambda }_1.$$

Make it simultaneous with the equation that the right end of the state equation is equal to 0, and get the optimal breeding scale:

$$N^{*}=(r-\frac{C}{ep})\frac{K}{2r},$$

at the same time, the sale rate is

$${\alpha }^{*}=\frac{1}{2}(r+\frac{C}{ep}),$$

because $U=\{\alpha |0<\alpha <r\}$, so we get

$$\alpha =min\{max\{0,\frac{1}{2}(r+\frac{C}{ep})\},r\}.$$

□

3.2. Optimal Breeding Strategy under the Model without Price Adjustment

In this case, we consider that when the price changes, the livestock farmers do not adjust the scale of livestock with the price, and then formulate the optimal breeding strategy. Since prices fluctuate under the effect of supply and demand, it is necessary to control the sale rate of livestock and control the scale of breeding in order to obtain maximum profit and maintain market stability. Like same profit function (6), in the constant price model, this profit function is

$$J(\alpha ,N,p)=\underset{0}{\overset{T}{\int }}(ep(t)\alpha (t)N(t)-CN(t))dt,$$

(10)

satisfying the equation of state:

$$\left\{\begin{array}{c}\frac{dN}{dt}=rN(1-\frac{N}{K})-\alpha (t)N,\hfill \\ \frac{dp}{dt}=\varphi (D-bep-\alpha (t)N),\hfill \end{array}\right.$$

(11)

with initial conditions $N(0)\ge 0,p(0)\ge 0.$

The same goal is to find ${\alpha }^{*},N^{*},p^{*}$ to maximize the above profit function, namely:

$$J({\alpha }^{*},N^{*},p^{*})=max\{J(\alpha ,N,p)|\alpha \in U\},$$

(12)

where if $Kr-4D\le 0$, $U=\{\alpha |0<\alpha <r\}$, if $Kr-4D>0$, $U=\{\alpha |0<\alpha <\frac{rK-\sqrt{Kr(Kr-4D)}}{2K},\frac{rK+\sqrt{Kr(Kr-4D)}}{2K}<\alpha <r\}$. U is determined by Theorem 2.

Theorem 8.

The optimal breeding formula is

$${\alpha }^{*}=\frac{\sqrt[3]{3}}{6K}m-\frac{r(-Kr+2D)\sqrt[3]{9}}{6m}+\frac{r}{2},N^{*}=\frac{K(r-{\alpha }^{*})}{r},p^{*}=\frac{K{{\alpha }^{*}}^2-K{\alpha }^{*}r+Dr}{ber},$$

where $m={\left[r[\sqrt{-3K^6{r}^4+18D{K}^5{r}^3-36D^2{K}^4{r}^2+81K^4{b}^2{c}^2+24D^3{K}^{3}r}+9cb{K}^2]\right]}^{\frac{1}{3}}.$

If ${\alpha }^{*}\in U$, the optimal strategy is

$$\alpha =\left\{\begin{array}{c}0\hfill \\ {\alpha }^{*}\hfill \\ r\hfill \end{array}\right.\begin{array}{c}N(0)<N^{*},\\ N(0)=N^{*},\\ N(0)>N^{*}.\end{array}$$

If ${\alpha }^{*}>{\alpha }_{max}=r$, the optimal strategy is to use the maximum sale rate. If ${\alpha }^{*}<0$, the optimal strategy is not to sell livestock.

Proof. 

According to the maximization principle [31], the Hamiltonian is constructed as

$$\begin{array}{c}H=(ep\alpha N-CN)+{\lambda }_1\left[rN(1-\frac{N}{K})-\alpha N\right]+{\lambda }_2\left[D-bep-\alpha N\right]\varphi \hfill \\ =(ep-{\lambda }_1-\varphi {\lambda }_2)\alpha N-CN+{\lambda }_{1}rN(1-\frac{N}{K})+{\lambda }_2\varphi (D-bep),\hfill \end{array}$$

where ${\lambda }_1,{\lambda }_2$ are adjoint variables. They satisfy

$$\frac{\partial {\lambda }_1}{\partial t}=-\frac{\partial H}{\partial N}=C-{\lambda }_{1}r+{\lambda }_1\frac{2rN}{K},$$

$${\lambda }_1(T)=0,$$

$$\frac{\partial {\lambda }_2}{\partial t}=-\frac{\partial H}{\partial p}=-e\alpha N+\varphi {\lambda }_{2}be,$$

$${\lambda }_2(T)=0.$$

Obviously, H is linear with respect to the control variable $\alpha$, and the control is of bang–bang type [32], so the coefficient of H with respect to $\alpha$ is defined as

$$\sigma (N,t)=(ep-{\lambda }_1-\varphi {\lambda }_2)N,$$

To maximize H, then,

$$\alpha =\left\{\begin{array}{c}0\hfill \\ uncertain\hfill \\ r\hfill \end{array}\right.\begin{array}{c}\sigma (N,t)<0,\\ \sigma (N,t)=0,\\ \sigma (N,t)>0.\end{array}$$

The optimal control solution under uncertainty is discussed. If $\sigma (N,t)=0$,

$$\frac{\partial H}{\partial \alpha }=\sigma (N,t)=0,$$

$$ep={\lambda }_1+\varphi {\lambda }_2.$$

Due to the influence of linear control, the final state of the system is

$$\left\{\begin{array}{c}\frac{dN}{dt}=0,\hfill \\ \frac{dp}{dt}=0,\hfill \\ \frac{\partial {\lambda }_i}{\partial t}=0.\hfill \end{array}\right.$$

Thus, the optimal control variable satisfies

$$\left\{\begin{array}{c}e{p}^{*}={\lambda }_1+\varphi {\lambda }_2,\hfill \\ {\alpha }^{*}{N}^{*}=r{N}^{*}(1-\frac{N^{*}}{K}),\hfill \\ {\alpha }^{*}{N}^{*}=D-be{p}^{*},\hfill \\ \frac{\partial {\lambda }_1}{\partial t}=C-{\lambda }_{1}r+{\lambda }_1\frac{2r{N}^{*}}{K}=0,\hfill \\ \frac{\partial {\lambda }_2}{\partial t}=-e\alpha N^{*}+\varphi {\lambda }_{2}be=0.\hfill \end{array}\right.$$

The equation for solving ${\alpha }^{*}$ can be obtained from the above equation:

$$-4K{\alpha }^3+6Kr{\alpha }^2-2K{r}^2\alpha -2Dr\alpha +cbr+D{r}^2=0;$$

one real root and two complex roots are obtained from the solving process, which is conducted by using Maple software, but two complex roots are meaningless. Therefore, the optimal breeding result is

$$\left\{\begin{array}{c}{\alpha }^{*}=\frac{-3K^2{r}^2+6DKr+3Kr{m}_1-m_1^2}{6K{m}_1},\hfill \\ N^{*}=\frac{K(r-{\alpha }^{*})}{r},\hfill \\ p^{*}=\frac{K{{\alpha }^{*}}^2-K{\alpha }^{*}r+Dr}{ber},\hfill \end{array}\right.$$

(13)

where the set intermediate quantity $m_i(i=1,2)$ satisfies

$$\left\{\begin{array}{c}{m}_1={(3Kr{m}_2-27C{K}^{2}br)}^{\frac{1}{3}},\hfill \\ m_2={(3K(-K^3{r}^4+6D{K}^2{r}^3+27C^{2}K{b}^2-12D^{2}K{r}^2+8D^{3}r))}^{\frac{1}{2}}.\hfill \end{array}\right.$$

If ${\alpha }^{*}\in U$, the optimal breeding results (13) is effective. □

3.3. Optimal Breeding Strategy under the Model with Price Adjustment

In the model (4), we have assumed that the direct effect of price on the livestock scale is $\epsilon e(p-p_0)$. Since we consider whether $\epsilon e(p-p_0)$ incur costs, there are two kinds of profit functions. If breeding is the mode of self-sufficiency, which means that livestock farms regulate their own productivity to achieve $\epsilon e(p-p_0)$, the profit function does not consider the cost of $\epsilon e(p-p_0)$. If breeding is the mode of outsourcing young livestock, which means that the livestock farm needs to buy or sell young livestock from other livestock farms to achieve $\epsilon e(p-p_0)$, the profit function needs to consider the cost of $\epsilon e(p-p_0)$. In the following, the optimal breeding strategies of the two modes are obtained.

3.3.1. The Mode of Self-Sufficiency

In the mode of self-sufficiency, farmers want to make higher profits by regulating productivity with prices, $\epsilon$ is determined by the productivity of the farm itself and income increases with $\epsilon$, which is explained specifically in Section 4.2. Therefore, $\epsilon$ is not available as a control variable; only the sale rate $\alpha$ is used as the control variable. $M(t)=0$, and the profit function is

$$J(\alpha ,N,p)=\underset{0}{\overset{T}{\int }}(ep(t)\alpha (t)N(t)-CN(t))dt.$$

(14)

It satisfies the equation of state:

$$\left\{\begin{array}{c}\frac{dN}{dt}=rN(1-\frac{N}{K})+\epsilon e(p-p_0)-\alpha (t)N,\hfill \\ \frac{dp}{dt}=\varphi (D-bep-\alpha (t)N),\hfill \end{array}\right.$$

(15)

with initial conditions $N(0)\ge 0,p(0)\ge 0.$

The same goal is to find ${\alpha }^{*},N^{*},p^{*}$ to maximize the above profit function, namely:

$$J({\alpha }^{*},N^{*},p^{*})=max\{J(\alpha ,N,p)|\alpha \in U\},$$

(16)

where

$$U=\{\alpha |2Drb-\alpha K(rb-\epsilon \alpha -b\alpha +\sqrt{{(rb-\epsilon \alpha -b\alpha )}^2+4\frac{rb\epsilon }{K}(D-be{p}_0)})>0,\alpha >0\}.$$

U is determined by Theorem 5.

Theorem 9.

The optimal control formula is

$${\alpha }^{*}=\frac{-Kbe{p}_0\epsilon +K{N}^{*}br-{N^{*}}^{2}br+DK\epsilon }{K{N}^{*}(b+\epsilon )},N^{*}=m_1+\frac{K}{2},$$

$$p^{*}=\frac{Ke{p}_0\epsilon -K{N}^{*}r+{N^{*}}^{2}r+DK}{Ke(b+\epsilon )},$$

where $m_i(i=1,\dots ,6)$ satisfies

$$\left\{\begin{array}{c}{m}_1=\frac{m_2}{6br}-\frac{K(m_6-Kbr+2Db-2D\epsilon )}{2m_2},\hfill \\ m_2={(3K{b}^{2}r({(\frac{3K(m_3+m_4+m_5+m_6)}{b})}^{\frac{1}{2}}-9CK{b}^2-18CKb\epsilon -9CK{\epsilon }^2))}^{\frac{1}{3}},\hfill \\ m_3=K^2{b}^2{r}^3(-Kbr+6Db-6D\epsilon )+27C^{2}Kb{(b+\epsilon )}^4,\hfill \\ m_4=4D^{2}r{(b-\epsilon )}^2(-3Kbr+2Db-2D\epsilon ),\hfill \\ m_5=m_6(m_6(m_6-3Kbr+6Db-6D\epsilon )-3Kbr(-Kbr+4Db-4D\epsilon )+12D^2({(b-\epsilon )}^2+{\epsilon }^2)),\hfill \\ m_6=4be{p}_{0}r\epsilon .\hfill \end{array}\right.$$

If ${\alpha }^{*}\in U$, the optimal strategy is

$$\alpha =\left\{\begin{array}{c}0\hfill \\ {\alpha }^{*}\hfill \\ {\alpha }_{max}\hfill \end{array}\right.\begin{array}{c}N(0)<N^{*},\\ N(0)=N^{*},\\ N(0)>N^{*}.\end{array}$$

If ${\alpha }^{*}>{\alpha }_{max}$, the optimal strategy is to use the maximum sale rate. ${\alpha }_{max}$ is decided by U. If ${\alpha }^{*}<0$, the optimal strategy is not to sale livestock.

Proof. 

According to the maximization principle [31], the Hamiltonian is constructed as

$$\begin{array}{c}H=(ep\alpha N-CN)+{\lambda }_1\left[rN(1-\frac{N}{K})+\epsilon e(p-p_0)-\alpha N\right]+{\lambda }_2\left[D-bep-\alpha N\right]\varphi \hfill \\ =(ep-{\lambda }_1-\varphi {\lambda }_2)\alpha N-CN+{\lambda }_{1}rN(1-\frac{N}{K})+{\lambda }_1\epsilon e(p-p_0)+{\lambda }_2\varphi (D-bep),\hfill \end{array}$$

where ${\lambda }_1,{\lambda }_2$ are adjoint variables. They satisfy

$$\frac{\partial {\lambda }_1}{\partial t}=-\frac{\partial H}{\partial N}=C-{\lambda }_{1}r+{\lambda }_1\frac{2rN}{K},$$

$${\lambda }_1(T)=0,$$

$$\frac{\partial {\lambda }_2}{\partial t}=-\frac{\partial H}{\partial p}=-e\alpha N+{\lambda }_2\varphi be-{\lambda }_1\epsilon e,$$

$${\lambda }_2(T)=0.$$

Obviously, H is linear with respect to the control variable $\alpha$, and the control is of bang–bang type [32], so the coefficient of H with respect to $\alpha$ is defined as

$$\sigma (N,t)=(ep-{\lambda }_1-\varphi {\lambda }_2)N,$$

To maximize H, then,

$$\alpha =\left\{\begin{array}{c}0\hfill \\ uncertain\hfill \\ {\alpha }_{max}\hfill \end{array}\right.\begin{array}{c}\sigma (N,t)<0,\\ \sigma (N,t)=0,\\ \sigma (N,t)>0.\end{array}$$

The optimal control solution under uncertainty is discussed. If $\sigma (N,t)=0$,

$$\frac{\partial H}{\partial \alpha }=\sigma (N,t)=0,$$

$$ep={\lambda }_1+\varphi {\lambda }_2.$$

Due to the influence of linear control, the final state of the system is

$$\left\{\begin{array}{c}\frac{dN}{dt}=0,\hfill \\ \frac{dp}{dt}=0,\hfill \\ \frac{\partial {\lambda }_i}{\partial t}=0.\hfill \end{array}\right.$$

Thus, the optimal control variable satisfies

$$\left\{\begin{array}{c}e{p}^{*}={\lambda }_1+\varphi {\lambda }_2,\hfill \\ {\alpha }^{*}{N}^{*}=r{N}^{*}(1-\frac{N^{*}}{K})+{\epsilon }^{*}e(p^{*}-p_0),\hfill \\ {\alpha }^{*}{N}^{*}=D-be{p}^{*},\hfill \\ \frac{\partial {\lambda }_1}{\partial t}=C-{\lambda }_{1}r+{\lambda }_1\frac{2r{N}^{*}}{K}=0,\hfill \\ \frac{\partial {\lambda }_2}{\partial t}=-e{\alpha }^{*}{N}^{*}+\varphi {\lambda }_{2}be-{\lambda }_1\epsilon e=0.\hfill \end{array}\right.$$

From the above equations, we can get the optimal breeding result:

$$\left\{\begin{array}{c}{N}^{*}=m_1+\frac{K}{2},\hfill \\ {\alpha }^{*}=\frac{-Kbe{p}_0\epsilon +K{N}^{*}br-{N^{*}}^{2}br+DK\epsilon }{K{N}^{*}(b+\epsilon )},\hfill \\ p^{*}=\frac{Ke{p}_0\epsilon -K{N}^{*}r+{N^{*}}^{2}r+DK}{Ke(b+\epsilon )},\hfill \end{array}\right.$$

(17)

where the set intermediate quantity $m_i(i=1,\dots ,6)$ satisfies

$$\left\{\begin{array}{c}{m}_1=\frac{m_2}{6br}-\frac{K(m_6-Kbr+2Db-2D\epsilon )}{2m_2},\hfill \\ m_2={(3K{b}^{2}r({(\frac{3K(m_3+m_4+m_5+m_6)}{b})}^{\frac{1}{2}}-9CK{b}^2-18CKb\epsilon -9CK{\epsilon }^2))}^{\frac{1}{3}},\hfill \\ m_3=K^2{b}^2{r}^3(-Kbr+6Db-6D\epsilon )+27C^{2}Kb{(b+\epsilon )}^4,\hfill \\ m_4=4D^{2}r{(b-\epsilon )}^2(-3Kbr+2Db-2D\epsilon ),\hfill \\ m_5=m_6(m_6(m_6-3Kbr+6Db-6D\epsilon )-3Kbr(-Kbr+4Db-4D\epsilon )+12D^2({(b-\epsilon )}^2+{\epsilon }^2)),\hfill \\ m_6=4be{p}_{0}r\epsilon .\hfill \end{array}\right.$$

If ${\alpha }^{*}\in U$, the optimal breeding results (17) is effective. □

3.3.2. The Mode of Outsourcing Young Livestock

In the mode of outsourcing young livestock, livestock farms need to buy and sell young livestock, because the productivity of farms is not enough to adjust the scale. The buying and selling cost of young livestock is $C_1\epsilon e(p-p_0)$, where $C_1$ is the price of each young animal. $\epsilon$ and $\alpha$ are all considered as control variables. The livestock farmers hope to adjust the scale to maximize profits through the changes of prices, but they may have to spend some money, and they need to decide whether to adjust or not, so the adjustment coefficient $\epsilon$ should be used as the control variable. $M(t)=C_1\epsilon e(p-p_0)$, and the payment function is

$$J(\alpha ,\epsilon ,N,p)=\underset{0}{\overset{T}{\int }}(ep(t)\alpha (t)N(t)-CN(t)-C_1\epsilon (t)e(p(t)-p_0))dt,$$

(18)

satisfying the equation of state:

$$\left\{\begin{array}{c}\frac{dN}{dt}=rN(1-\frac{N}{K})+\epsilon (t)e(p-p_0)-\alpha (t)N,\hfill \\ \frac{dp}{dt}=\varphi (D-bep-\alpha (t)N),\hfill \end{array}\right.$$

(19)

with initial conditions $N(0)\ge 0,p(0)\ge 0.$

In this mode, the goal is to find a set of $({\alpha }^{*},{\epsilon }^{*},N^{*},p^{*})$ to maximize the above profit function, namely,

$$J({\alpha }^{*},{\epsilon }^{*},N^{*},p^{*})=max\{J(\alpha ,\epsilon ,N,p)|(\alpha ,\epsilon )\in U\},$$

(20)

where

$$U=\{(\alpha ,\epsilon )|K(rb-\epsilon \alpha -b\alpha +\sqrt{{(rb-\epsilon \alpha -b\alpha )}^2+4\frac{rb\epsilon }{K}(D-be{p}_0)})>0,\alpha >0,\epsilon >0\}.$$

Theorem 10.

The optimal control formula is

$${\alpha }^{*}=\frac{(D-b{C}_1)r{C}_1}{K(r{C}_1-C)},{\epsilon }^{*}=\frac{b\left[2{C_1}^{2}r(r{C}_1-C)-2C_{1}rK(r{C}_1-C)+K{(r{C}_1-C)}^2\right]}{2{C_1}^{2}r(D+b{C}_1-2br{p}_0)},$$

$$N^{*}=\frac{K(r{C}_1-C)}{2C_{1}r},p^{*}=\frac{D+b{C}_1}{2be}.$$

If $({\alpha }^{*},{\epsilon }^{*})\in U$, the optimal strategy is

$$\alpha =\left\{\begin{array}{c}0\hfill \\ {\alpha }^{*}\hfill \\ {\alpha }_{max}\hfill \end{array}\right.\begin{array}{c}N(0)<N^{*},\\ N(0)=N^{*},\\ N(0)>N^{*},\end{array}$$

$$\epsilon =\left\{\begin{array}{c}0\hfill \\ {\epsilon }^{*}\hfill \\ {\epsilon }_{max}\hfill \end{array}\right.\begin{array}{c}p(0)<p^{*},\\ p(0)=p^{*},\\ p(0)>p^{*},\end{array}$$

where $({\alpha }_{max},{\epsilon }_{max})$ is decided by U. If ${\epsilon }^{*}<0$, the optimal strategy is $\epsilon =0$ and the optimal result is the result (13).

Proof. 

According to the maximization principle [31], the Hamiltonian is constructed as

$$\begin{array}{c}H=(ep\alpha N-CN-C_{1}e\epsilon (p-p_0))+{\lambda }_1\left[rN(1-\frac{N}{K})+\epsilon e(p-p_0)-\alpha N\right]+{\lambda }_2\left[D-bep-\alpha N\right]\varphi \hfill \\ =(ep-{\lambda }_1-\varphi {\lambda }_2)\alpha N+({\lambda }_1-C_1)e(p-p_0)\epsilon -CN+{\lambda }_{1}rN(1-\frac{N}{K})+{\lambda }_2\varphi (D-bep),\hfill \end{array}$$

where ${\lambda }_1,{\lambda }_2$ are adjoint variables. They satisfy

$$\frac{\partial {\lambda }_1}{\partial t}=-\frac{\partial H}{\partial N}=C-{\lambda }_{1}r+{\lambda }_1\frac{2rN}{K},$$

$${\lambda }_1(T)=0,$$

$$\frac{\partial {\lambda }_2}{\partial t}=-\frac{\partial H}{\partial p}=-e\alpha N+{\lambda }_2\varphi be-{\lambda }_1\epsilon e+C_{1}e\epsilon ,$$

$${\lambda }_2(T)=0.$$

Obviously, H is linear with respect to the control variable $\alpha$ and $\epsilon$, and the control is of bang–bang type [32], so the coefficient of H with respect to $\alpha$ and $\epsilon$ is defined as

$${\sigma }_1(N,t)=(ep-{\lambda }_1-\varphi {\lambda }_2)N,$$

$${\sigma }_2(p,t)=({\lambda }_1-C_1)e(p-p_0).$$

To maximize H, then,

$$\alpha =\left\{\begin{array}{c}0\hfill \\ uncertain\hfill \\ {\alpha }_{max}\hfill \end{array}\right.\begin{array}{c}{\sigma }_1(N,t)<0,\\ {\sigma }_1(N,t)=0,\\ {\sigma }_1(N,t)>0,\end{array}$$

$$\epsilon =\left\{\begin{array}{c}0\hfill \\ uncertain\hfill \\ {\epsilon }_{max}\hfill \end{array}\right.\begin{array}{c}{\sigma }_2(p,t)<0,\\ {\sigma }_2(p,t)=0,\\ {\sigma }_2(p,t)>0.\end{array}$$

The optimal control solution under uncertainty is discussed. If ${\sigma }_1(N,t)=0$, ${\sigma }_2(p,t)=0$,

$$\frac{\partial H}{\partial \alpha }=\sigma (N,t)=0,$$

$$ep={\lambda }_1+\varphi {\lambda }_2,$$

$$\frac{\partial H}{\partial \epsilon }={\sigma }_2(p,t)=0,$$

$$C_1={\lambda }_1.$$

Due to the influence of linear control, the final state of the system is

$$\left\{\begin{array}{c}\frac{dN}{dt}=0,\hfill \\ \frac{dp}{dt}=0,\hfill \\ \frac{\partial {\lambda }_i}{\partial t}=0.\hfill \end{array}\right.$$

Thus, the optimal control variable satisfies

$$\left\{\begin{array}{c}e{p}^{*}={\lambda }_1+\varphi {\lambda }_2,\hfill \\ C_1={\lambda }_1,\hfill \\ {\alpha }^{*}{N}^{*}=r{N}^{*}(1-\frac{N^{*}}{K})+{\epsilon }^{*}e(p^{*}-p_0),\hfill \\ {\alpha }^{*}{N}^{*}=D-be{p}^{*},\hfill \\ \frac{\partial {\lambda }_1}{\partial t}=C-{\lambda }_{1}r+{\lambda }_1\frac{2r{N}^{*}}{K}=0,\hfill \\ \frac{\partial {\lambda }_2}{\partial t}=-e{\alpha }^{*}{N}^{*}+{\lambda }_2\varphi be=0.\hfill \end{array}\right.$$

From the above equation, the optimal breeding result can be obtained as

$$\left\{\begin{array}{c}{N}^{*}=\frac{K(r{C}_1-C)}{2C_{1}r},\hfill \\ p^{*}=\frac{D+b{C}_1}{2be},\hfill \\ {\alpha }^{*}=\frac{(D-b{C}_1)r{C}_1}{K(r{C}_1-C)},\hfill \\ {\epsilon }^{*}=\frac{b\left[2{C_1}^{2}r(r{C}_1-C)-2C_{1}rK(r{C}_1-C)+K{(r{C}_1-C)}^2\right]}{2{C_1}^{2}r(D+b{C}_1-2br{p}_0)}.\hfill \end{array}\right.$$

(21)

If the above-mentioned $({\alpha }^{*},{\epsilon }^{*})$ is within the control set of U, the optimal breeding results (21) are effective. □

4. Numerical Analysis

In Section 3, we got the theoretical results of the optimal breeding strategy under different modes, but we need to verify and understand the existence of control solutions more intuitively, and there we did not know which strategies should be adopted to obtain the maximum profit in the instance. Therefore, in this section, in order to have a more comprehensive understanding of the breeding strategy, we report simulations for the two cases where demand is greater than supply and demand is less than supply. Then, we graphically show the effects of control variables on profits. Finally, the optimal strategy results are given for each model in both cases. Through the comparative analysis of these optimal results, we can obtain the most effective optimal breeding strategy for maintaining the supply–demand balance and obtaining the maximum profit.

4.1. Solutions of Models

In order to be able to simulate two cases where demand is less than supply and demand is greater than supply, we set two groups of parameter values to present these two cases, which are given in

Table 1. Parameter description and two groups of parameter values.

Parameter	Description	The First Group	The Second Group
r	Intrinsic growth rate	0.7	0.7
K	Maximum environmental capacity	$2\times {10}^6$	$1\times {10}^6$
D	Maximum demand	$8\times {10}^5$	$8\times {10}^5$
b	Correlation coefficient of the influence of demand on price	350	350
e	The average weight of livestock put on the market	65	65
$p_0$	The balanced price determined by the farm	20	20
$\epsilon$	Correlation coefficient of livestock scale affected by the price	100	100
$\alpha$	The sale rate	0.05	0.2
C	Breeding cost per unit time	10	10
$C_1$	The cost of buying or selling each young livestock	300	300
$N(0)$	The total number of livestock at initial moment	$5\times {10}^4$	$1.5\times {10}^6$
$p(0)$	Initial moment price	35	22
$\varphi$	Proportionality constant	1	1

. Mark $N_i(t)(i=1,2,3,4)$ and $p_i(i=1,2,3,4)$ as N and p of the systems (7), (11), (15) and (19), in order. $R_i(t)$ is the J of models (6), (10), (14) and (18). $R_i$, $N_i$ and $p_i$ correspond to each other.

Based on two groups of parameter values in Table 1, Figure 2, Figure 3 and Figure 4 depict the the changes in price, livestock quantity and profits, and we set $T=40$ month. First of all, let us look at the price changes. $p_1$ remains unchanged because it is not affected by supply and demand. In Figure 2a, $p_2$, $p_3$ and $p_4$ decrease, which indicates that demand is less than supply according to model (2). When the price falls, the demand increases until it equals the supply, and the price tends to be stable. In contrast, in Figure 2b, $p_2$, $p_3$ and $p_4$ rise, which indicates that demand is greater than supply. As shown in Figure 2, $p_3$ and $p_4$ reach equilibrium earlier than $p_2$, so we can conclude that $\epsilon e(p-p_0)$ helps to adjust the balance of supply and demand more quickly. Besides, why is $p_3(t)=p_4(t)<p_2(t)$? This is explained when introducing Figure 3 in the following.

For livestock quantity, since both $N_1$ and $N_2$ are not affected by the price, they have the same trajectory. Both $N_3$ and $N_4$ are affected by the price, so their trajectory is the same. Figure 3a shows that when demand is less than supply, it means that the supply is large at this time, so the amount of livestock increases. For the reason that $N_3$ and $N_4$ are affected by the regulation $\epsilon e(p-p_0)$ and $\epsilon e(p-p_0)>0$, there is $N_3(t)=N_4(t)>N_2(t)$, which means that $\alpha N_3(t)=\alpha N_4(t)>\alpha N_2(t)$. Therefore, there is $p_3(t)=p_4(t)<p_2(t)$ according to the model (2). Figure 3b shows that when demand is greater than supply, it means that the supply is less at this time, so the amount of livestock decreases. In addition due to the effect of $\epsilon e(p-p_0)>0$, there is $N_3(t)=N_4(t)>N_2(t)$ and $p_3(t)=p_4(t)<p_2(t)$. As shown in Figure 3, $N_3$ and $N_4$ reach equilibrium earlier than $N_2$, so we can also conclude that $\epsilon e(p-p_0)$ helps to adjust the balance of supply and demand more quickly.

As shown in Figure 4, whether demand is less than supply or demand is greater than supply, when ignoring $R_1$, $R_3$ is the maximum at any moment, which indicates that regulating one’s own productivity can be more profitable than importing young livestock from other pastures and not regulating with prices. In Figure 4a, $R_4(t)>R_2(t)$ before $t=22$, but after $t=22$ $R_4(t)<R_2(t)$ and $R_2(40)>R_4(40)$. However, in Figure 4b, there is $R_4(t)>R_2(t)$ at any moment, and total profits are $R_4(40)>R_2(40)$. This shows that importing young livestock from other farms does not always increase profits, which depends on the cost of young livestock.

4.2. The Effect of Control Variables on Profits

To show that the optimal solution for the systems (8), (12), (16) and (20) exists and is reasonable, we analyze the effect of the sale rate on profits in each model and the effect of the adjustment rate on profits of models (14) and (18).

As shown in Figure 5, with the adjustment of the sale rate $\alpha$ within a reasonable range, a maximum value of profits occurs for each model, which indicates that revenue can be maximized by adjusting the sale rate $\alpha$. As shown in Figure 6a, in the mode of self-sufficiency, the profit increases with the increase of $\epsilon$, and there is no maximum profit, so $\epsilon$ should not be regarded as a control variable. In addition, this shows that livestock farmers can appropriately increase their production capacity to obtain higher profits. From Figure 6b, it can be seen that in the mode of outsourcing young livestock, there is a maximum profit through the change of $\epsilon$, so $\epsilon$ should be considered as a control variable in this mode. Through sensitivity analysis, we can draw the conclusion that adjusting the control variables can get the optimal breeding strategy of each mode, which is reasonable and effective.

4.3. Optimal Breeding Strategy

In Section 3, we only gave the theoretical solution with optimal strategy in each mode. Although we could realize the balance between supply and demand, we did not know which optimal strategy can get the maximum profit. In this section, we study how to choose the optimal strategy under specific cases.

The optimal breeding strategy under each mode was simulated by applying the two groups of parameters in Table 1, and the results of four stratgies are presented in

Table 2. Optimal breeding results based on the first group of parameter values. t is the start time of selling. Since $N(0)<N^{*}$, the optimal strategy is to sell when the livestock scale grows near $N^{*}$. $R^{*}$ is the optimal profit obtained from 0 to terminal time $T=40$. R is the profit obtained from 0 to terminal time $T=40$ under no control. $\epsilon =100$ is determined by the productivity of the farm itself rather than by optimal control.

Model	${\mathit{\alpha }}^{*}$	${\mathit{\epsilon }}^{*}$	${\mathit{N}}^{*}$	${\mathit{p}}^{*}$	t	${\mathit{R}}^{*}$	R
(6) and (7)	0.3538	∖	989,142	35	5th month	$2.811\times {10}^{10}$	$6.729\times {10}^9$
(10) and (11)	0.3672	∖	950,814	19.82	5th month	$1.616\times {10}^{10}$	$5.94\times {10}^9$
(14) and (15)	0.3707	100	938,373	19.88	5th month	$1.708\times {10}^{10}$	$6.580\times {10}^9$
(18) and (19)	0.3648	238.89	952,587	19.89	5th month	$1.609\times {10}^{10}$	$5.708\times {10}^9$

and

Table 3. Optimal breeding results based on the second group of parameter values. $R(0)$ is the profit earned at the initial moment, Since $N(0)>N^{*}$, every optimal strategy is to sell the excess over $N^{*}$ and the profit is $R(0)$. The meanings of $R^{*}$, R and $\epsilon =100$ are the same as in Table 2.

Model	${\mathit{\alpha }}^{*}$	${\mathit{\epsilon }}^{*}$	${\mathit{N}}^{*}$	${\mathit{p}}^{*}$	$\mathit{R}(0)$	${\mathit{R}}^{*}$	R
(6) and (7)	0.3538	∖	494,571	22	1,437,763,470	$1.125\times {10}^{10}$	$8.088\times {10}^9$
(10) and (11)	0.3539	∖	494,428	27.47	1,437,967,960	$1.374\times {10}^{10}$	$1.064\times {10}^{10}$
(14) and (15)	0.4329	100	491,424	25.81	1,442,263,680	$1.552\times {10}^{10}$	$1.158\times {10}^{10}$
(18) and (19)	0.3539	∖	494,428	27.47	1,437,967,960	$1.374\times {10}^{10}$	$1.096\times {10}^{10}$

, and the optimal profit results are presented in Figure 7 and Figure 8. As shown by the results, according to the system (6) and (7) when the price is constant, we only need to adjust the slaughter rate $\alpha$ and maintain the optimal breeding scale $N^{*}$. However, when the supply and demand are unbalanced, we need to compare the optimal profits under the other three optimal strategies. Based on the first group of parameters, the values of $R^{*}$ in Table 2 are compared. We finally choose the optimal breeding strategy of systems (14) and (15), and adopt a behavior that reducing the productivity of the farm because of $p^{*}<p_0$. The sub-optimal strategy is to sell some young livestock. The results show that adjusting the livestock scale with price can be more profitable than not adjusting.

Once more in case 2, according to the results in Table 3 we choose the optimal strategy results of systems (14) and (15), and adopt a behavior that increasing the productivity of own farms because of $p^{*}>p_0$. When computing the optimal result of the system (18) and (19), it appears that $\epsilon <0$, so the strategy (21) does not hold, and the final strategy is the same as the system (10) and (11). Based on the optimal strategy results, we can conclude that proper regulation of farms productivity can be more profitable than buying and selling young livestock.

Our model has some drawbacks, when we adopt the optimal breeding strategy, as shown in Figure 8, there is an sudden change in the price, which in practice is impossible and is not allowed to happen, because it is not conducive to market stability. This phenomenon occurs because we consider in the model that livestock are directly circulated to the market after they are sold, while in fact, in order to regulate the stability of the market, the government stocks a certain amount of the products of livestock. Therefore, when applying the optimal breeding strategy, it is also necessary for the government to play the role of stabilizing the market to make the price change gradually instead of changing abruptly.

5. Conclusions

The main problems of the livestock market in China are how to develop production, promote market balance and ensure consumer demand [7]. In order to achieve the balance of supply and demand and maximize the profits of breeding, we studied the livestock model under dynamic pricing and found the optimal breeding strategy. First of all, we set up a price model regulated by supply and demand. Combined with the price model and behaviors of farmers, we built a model of livestock not directly affected by price and a model of livestock directly affected by price, where the effect of price on the livestock is $\epsilon e(p-p_0)$, which is caused by livestock farmers regulating their farm productivity or buying and selling young livestock. We performed a simple analysis of the models, obtaining the conditions for the existence of a positive equilibrium, and discussed the stability of that equilibrium.

Secondly, according to the revenue and expenditure of each model, we established four objective profit functions. We set $\alpha$ and $\epsilon$ as control variables and obtained the necessary conditions for the optimal control solutions of the four systems by applying the Pontryagin maximization principle. Further, in order to explain the existence of the optimal breeding strategy of each system, we graphically showed the effects of $\alpha$ and $\epsilon$ on profits. The results show that the profit can be maximized by adjusting $\alpha$ and $\epsilon$.

Finally, to show that which optimal strategies should be chosen in practical applications, we set two groups of parameters to simulate the changes in livestock and price for two cases: demand less than supply and demand more than supply. The optimal strategies results were calculated for four modes in both cases, and the optimal strategy was found by comparing the profit under four optimal breeding strategies. The analysis of the optimal breeding strategy of the examples led to these conclusions:

• When the price is constant, if supply and demand do not affect the price at this time, the optimal control strategy under the constant price model can be used, but if they have an effect, this strategy is not reasonable;
• The effect of price on the livestock (i.e., $\epsilon e(p-p_0)$) can promote a faster balance between supply and demand;
• To get more profits, the livestock farmers should adjust the farm productivity reasonably, which is a key factor in breeding; The methods of adjusting the farm productivity include adjusting the proportion of female livestock, adjusting the breeding capacity of livestock, adjusting the construction of forage production base and adjusting the industrial support [7];
• For a specific instance, it is necessary to calculate the optimal strategy results under different behavior patterns, so we can go on to choose the most suitable strategy by comparing the optimal profits.

To sum up, the above points are the findings obtained during our research, and the most important result of this paper is that we have got four strategies to maximize profits and balance supply and demand under different prices and different behaviors of farmers, namely, Theorems 7–10. For a specific problem, we can calculate the profit results of the four strategies. Therefore, for farmers, they need to integrate their personal situations and compare the profits under the four strategies, i.e., (8), (12), (16) and (20), so as to ensure that the adopted strategy is the best.

Through research, we solved how to choose the optimal breeding strategy and adopt the most effective method to balance the supply and demand of the market and maximize the profits of livestock farmers. Next, we should consider infectious diseases, dynamic costs and other factors in the model, which will have a certain impact on profits and livestock scale, and further affect the optimal breeding strategy.