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# 数学代写|数学建模代写Mathematical Modeling代考|On the origin of nonlinearity

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## 数学代写|数学建模代写Mathematical Modeling代考|On the origin of nonlinearity

On the origin of nonlinearity. As was mentioned in section 1.5 , linear models obey the principle of superimposition. In this case, obtaining partial solutions and summarizing them, it is generally possible to construct a general solution (typical examples – formula (6), section 2 and the formula for the general solution of equation (3), section 4 in models of oscillations).
For nonlinear models the principle of superimposition is inapplicable, and the general solution can be found only in rare cases. Individual partial solutions of nonlinear equations may not reflect the character of the behavior of objects in a more general situation.

Nonlinearity can have many reasons. The fundamental laws of nature – the law of gravitation and the law of Coulomb – are nonlinear by origin (square-law relation of the force of interaction between masses or charges), and consequently the models based on them, generally speaking, are also nonlinear. Also contributing to the nonlinearity of models are the more complicated geometry of the phenomenon (see exercise 1 , section 4 and exercise 1 , section 5), various external actions (see equation (10) section 4) and, certainly, the change of the character of interaction in the object while changing its state (effect of saturation in models of populations, varying rigidity of the spring).

In essence only nonlinear models correspond to real phenomena, and linear models are valid only for descriptions of minor changes of parameters characterizing the object.

## 数学代写|数学建模代写Mathematical Modeling代考|Three regimes in a nonlinear model of population

Three regimes in a nonlinear model of population. As distinct from the Malthus model (10), section 1 , and model (12), section 1 , the coefficient of birth we shall consider dependent on the population $N(t)$, i.e. $\alpha=\alpha(N)$. The coefficient of mortality $\beta$ also depends on $N$. The equation of population dynamics
$$\frac{d N}{d t}=[\alpha(N)-\beta(N)] N$$
is nonlinear due to the variation of interaction characteristics within the population during the change of its state.

Assume for the sake of definiteness $\beta(N)=\beta_0=$ const, $\alpha(N)=\alpha_0(N)$, i.e. the birth rate is proportional to the population number (for example, because the population is interested in its own growth). Then the equation (1) will be transformed to
$$\frac{d N}{d t}=\alpha_0 N^2-\beta_0 N$$
with quadratic nonlinearity (peculiar also for some chemical reactions). Consider the behavior of function $N(t)$ at various initial populations $N(0)=N_0$ (Fig. 17).
a) At $N_0N_{\mathrm{cr}}$ the nature of the solution drastically changes as compared with cases a) and b): the number increases so quickly that it tends to infinity at finite time $t=t_f$. The value of $t_f$ is smaller, as $N_0$ increases

## 数学代写|数学建模代写Mathematical Modeling代考|Three regimes in a nonlinear model of population

$$\frac{d N}{d t}=[\alpha(N)-\beta(N)] N$$

$$\frac{d N}{d t}=\alpha_0 N^2-\beta_0 N$$

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