# 数学代写|数学建模代写Mathematical Modeling代考|Small oscillations at the interaction of two biological populations

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## 数学代写|数学建模代写Mathematical Modeling代考|Small oscillations at the interaction of two biological populations

Let two biological populations of numbers $N(t)$ and $M(t)$ coexist in the same territory, with the first ones being vegetarians, and the second ones being fed by representatives of the first population.

The rate of variation of $N(t)$ is determined by the first term on the right hand side of formula (10), section 1, describing the growth due to birth (effect of saturation is not taken into account; compare with (12), section 1) and from the rate of decrease due to the presence of the second population:
$$\frac{d N}{d t}=\left(\alpha_1-\beta_1 M\right) N$$
where $\alpha_1>0, \beta_1>0$, and the term $\beta_1 M N$ describes the enforced decrease (the natural mortality of population is neglected).

The greater the number of the first population, the faster the second population expands, while at its absence decreases with a rate proportional to the number $M(t)$ (thus its birth rate is not taken into account, as well as the effect of saturation):
$$\frac{d M}{d t}=\left(-\alpha_2+\beta_2 N\right) M$$
where $\alpha_2>0, \beta_2>0$.
Obviously, the system is in an equilibrium at $M_0=\alpha_1 / \beta_1$ and $N_0=$ $\alpha_2 / \beta_2$, when $d N / d t=d M / d t=0$. Consider small deviations of the system from the equilibrium values, i.e. represent the solution as $N=N_0+n, M=$ $M_0+m, n \ll N_0, m \ll M_0$. Substituting $N$ and $M$ into the equations (1), (2), we obtain (neglecting the terms of higher order of smallness)
\begin{aligned} & \frac{d n}{d t}=-\beta_1 N_0 m \ & \frac{d m}{d t}=-\beta_2 M_0 n . \end{aligned}

## 数学代写|数学建模代写Mathematical Modeling代考|Elementary model of variation of salary and employment

The trade market, where the employers and employees are interacting, is characterized by the salary $p(t)$ and occupation number $N(t)$. Let an equilibrium exist, i.e. a situation, when $N_0>0$ persons agree to work for salary $p_0>0$. If for any reasons this equilibrium is violated (for example, if some of the workers retire or the employers have financial difficulties), the functions $p(t)$ and $N(t)$ deviate from values $p_0, N_0$.

Consider that the employers change the salary proportionally to the deviation of number of employees from their equilibrium value. Then
$$\frac{d p}{d t}=-\alpha_1\left(N-N_0\right), \quad \alpha_1>0$$
Assume that the number of the workers also increases or decreases proportionally to the increase or decrease of the salary with respect the value $p_0$, i.e.
$$\frac{d N}{d t}=\alpha_2\left(p-p_0\right), \quad \alpha_2>0 .$$

Differentiating the first equation by $t$ and excluding $N$ from it with the help of the second equation, we come to a standard model of oscillation
$$\frac{d^2\left(p-p_0\right)}{d t^2}=\alpha_1 \alpha_2\left(p-p_0\right)$$
of the salary relative the equilibrium (analogously for $N(t)$ ). From the first integral of this equation
$$\alpha_1\left(N-N_0\right)^2+\alpha_2\left(p-p_0\right)^2=\text { const }>0$$

## 数学代写|数学建模代写MATHEMATICAL MODELING代考|SMALL OSCILLATIONS AT THE INTERACTION OF TWO BIOLOGICAL POPULATIONS

$$\frac{d N}{d t}=\left(\alpha_1-\beta_1 M\right) N$$

thusitsbirthrateisnottakenintoaccount, aswellastheef fectofsaturation:
$$\frac{d M}{d t}=\left(-\alpha_2+\beta_2 N\right) M$$

$$\frac{d n}{d t}=-\beta_1 N_0 m \quad \frac{d m}{d t}=-\beta_2 M_0 n$$

## 数学代写|数学建模代写MATHEMATICAL MODELING代考|ELEMENTARY MODEL OF VARIATION OF SALARY AND EMPLOYMENT

$$\frac{d p}{d t}=-\alpha_1\left(N-N_0\right), \quad \alpha_1>0$$

$$\frac{d N}{d t}=\alpha_2\left(p-p_0\right), \quad \alpha_2>0 .$$

$$\frac{d^2\left(p-p_0\right)}{d t^2}=\alpha_1 \alpha_2\left(p-p_0\right)$$

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