**9.3 The SIR (susceptible-infected-recovered) model**

The SIR is a simple mathematical model of epidemic [24], which divides the (fixed) population of *N* individuals into three" compartments" which may vary as a function of time *t*.

*S t*ð Þ are those susceptible but not yet infected with the disease,

*I t*ð Þ is the number of infectious individuals,

*R t*ð Þ are those individuals who have recovered from the disease and now have immunity to it,

the parameter *λ* describes the effective contact rate of the disease,

the parameter *μ* is the mean recovery rate.

The SIR model describes the change in the population of each of these compartments in terms of two parameters:

$$\begin{cases} \frac{dS(t)}{dt} = -\lambda \frac{S(t)I(t)}{N},\\ \frac{dI(t)}{dt} = \lambda \frac{S(t)I(t)}{N} - \mu I(t),\\ \frac{dR(t)}{dt} = \mu I(t), \end{cases} \tag{32}$$

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

and its restrictsion is

Let us consider a classical model

with an invariant *u t*, **y** � �.

*u t*ð Þ¼ , **x**ð Þ*t u*ð Þ¼ 0, **x**ð Þ 0 *C*:

**y**ð Þ¼ 0 **y**0*:*

8 ><

>:

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tions for PCP1:

function of time *t*.

immunity to it,

**74**

8 ><

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

>:

*dx*1ðÞ¼ *<sup>t</sup> <sup>a</sup>*1ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> dt* <sup>þ</sup> *<sup>b</sup>*1ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> <sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

*dx*2ðÞ¼ *<sup>t</sup> <sup>a</sup>*2ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> dt* <sup>þ</sup> *<sup>b</sup>*2ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> <sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

*dx*3ðÞ¼ *<sup>t</sup> <sup>a</sup>*3ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> dt* <sup>þ</sup> *<sup>b</sup>*3ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ*<sup>t</sup> <sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

Hence, the stochastic model has a representation

*dy*1ðÞ¼ *<sup>t</sup> <sup>a</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt* <sup>þ</sup> *<sup>b</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*d*wð Þþ*<sup>t</sup>* <sup>Ð</sup>

*dy*2ðÞ¼ *<sup>t</sup> <sup>a</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt* <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*d*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

*dy*3ðÞ¼ *<sup>t</sup> <sup>a</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt* <sup>þ</sup> *<sup>b</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*d*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

a differential equations system and having an invariant function.

*S t*ð Þ are those susceptible but not yet infected with the disease,

the parameter *λ* describes the effective contact rate of the disease,

*dS t*ð Þ *dt* ¼ �*<sup>λ</sup>*

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>>>>>>:

*dI t*ð Þ *dt* <sup>¼</sup> *<sup>λ</sup>*

*dR t*ð Þ

*dt* <sup>¼</sup> *<sup>μ</sup>I t*ð Þ,

**9.3 The SIR (susceptible-infected-recovered) model**

*I t*ð Þ is the number of infectious individuals,

the parameter *μ* is the mean recovery rate.

ments in terms of two parameters:

8 ><

>:

*dy*1ðÞ¼ *<sup>t</sup> <sup>F</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt*, *dy*2ðÞ¼ *<sup>t</sup> <sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt*, *dy*3ðÞ¼ *<sup>t</sup> <sup>F</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*dt*,

Further, we determine complementary function which is unit of control func-

*<sup>s</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � <sup>¼</sup> *<sup>a</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � � *<sup>F</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �, *<sup>s</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � � *<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �, *<sup>s</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � <sup>¼</sup> *<sup>a</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � � � *<sup>F</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* � �*:*

Finally, we have constructed stochastic analogue for classical model described by

The SIR is a simple mathematical model of epidemic [24], which divides the (fixed) population of *N* individuals into three" compartments" which may vary as a

*R t*ð Þ are those individuals who have recovered from the disease and now have

The SIR model describes the change in the population of each of these compart-

*S t*ð Þ*I t*ð Þ

*S t*ð Þ*I t*ð Þ *<sup>N</sup>* ,

*<sup>N</sup>* � *<sup>μ</sup>I t*ð Þ,

Then we construct the GSDE system, taking into account the equality

(28)

(29)

(30)

(31)

(32)

*g*1ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*2ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*3ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ :*

*<sup>g</sup>*<sup>1</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* , *<sup>γ</sup>* � �*ν*ð Þ *dt*, *<sup>d</sup><sup>γ</sup>* ,

*<sup>g</sup>*<sup>2</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* , *<sup>γ</sup>* � �*ν*ð Þ *dt*, *<sup>d</sup><sup>γ</sup>* ,

*<sup>g</sup>*<sup>3</sup> *<sup>t</sup>*, **<sup>y</sup>**ð Þ*<sup>t</sup>* , *<sup>γ</sup>* � �*ν*ð Þ *dt*, *<sup>d</sup><sup>γ</sup>* ,

$$I(t) + I(t) + R(t) = N.\tag{33}$$

Let the model with strong perturbation be

$$\begin{cases} dS(t) = \left(-\lambda \frac{S(t)I(t)}{N} + s\_1(t, S(t), I(t), R(t))\right) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_1(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \\\ dI(t) = \left(\lambda \frac{S(t)I(t)}{N} - \mu I(t) + s\_2(t, S(t), I(t), R(t))\right) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_2(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \\\ dR(t) = (\mu S(t) + s\_3(t, S(t), I(t), R(t))) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_3(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \end{cases} \tag{34}$$

and

$$u(t, \mathcal{S}(t), I(t), R(t)) = \mathcal{S}(t) + I(t) + R(t) - N \equiv \mathbf{0}.\tag{35}$$

Suppose that the function *u t*ð Þ¼ , *x*, *y*, *z x* þ *y* þ *z* � *N* is a first integral, *v t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* <sup>2</sup>*e*�*<sup>t</sup>* <sup>þ</sup> *<sup>x</sup>* and *h t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup> <sup>y</sup>* are complementary functions, and *q t*ð Þ¼ , *x*, *y*, *z x* is arbitrary function. The initial condition is: *x*ð Þ¼ 0 1, *y*ð Þ¼ 0 0, *z*ð Þ¼ 0 0. Then constructed differential equations system has the form

$$
\begin{bmatrix} dx(t) \\ dy(t) \\ dz(t) \end{bmatrix} = \begin{bmatrix} 2e^{-t} \\ 0 \\ -2e^{-t} \end{bmatrix} dt + \begin{bmatrix} 0 \\ \varkappa(t) \\ -\varkappa(t) \end{bmatrix} d\mathbf{w}(t) + \int\_{R\_{\gamma}} \begin{bmatrix} 0 \\ \varkappa(t)\gamma \\ -\varkappa(t)\gamma \end{bmatrix} \nu(dt, d\gamma). \tag{36}
$$

Let us simulate a numerical solution of Eg.(36), where *N* ¼ 1 (for example). **Figure 1** shows simulation for system without jumps, the **Figure 2** shows the processes with jumps.

**Figure 1.** *Numerical solution for Eq.(36) without jumps.*

• During the process, the environment does not change in favor of one species,

Let us note: *N*1ð Þ*t* is the number of prey, and *N*2ð Þ*t* is the number of some predator, *ε*1, *ε*2, *η*<sup>1</sup> and *η*<sup>2</sup> are positive real parameters describing the interaction of

The populations change through time according to the pair of equations:

*dN*1ðÞ¼ *t N*1ð Þ*t* ð Þ *ε*<sup>1</sup> � *η*1*N*2ð Þ*t dt*, *dN*2ðÞ¼� *t N*2ð Þ*t* ð Þ *ε*<sup>2</sup> � *η*2*N*1ð Þ*t dt:*

*<sup>η</sup>*2*N*1ð Þ*<sup>t</sup>* <sup>¼</sup> *CNε*<sup>1</sup>

<sup>2</sup> ð Þ*t e*

*Rγ*

*Rγ*

*<sup>η</sup>*2*x*1ð Þ*<sup>t</sup>* � *Cxε*<sup>1</sup>

Let us assume that *ε*<sup>1</sup> ¼ 2, *ε*<sup>2</sup> ¼ 1, *η*<sup>1</sup> ¼ *η*<sup>2</sup> ¼ 1, and *C* ¼ 1, and initial condition is

<sup>2</sup> ð Þ*t e*

�*z*2ð Þ *<sup>t</sup>*,*x*,*y*,*<sup>γ</sup> <sup>z</sup>*1ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup> <sup>z</sup>*2ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup>* ð Þ *<sup>z</sup>*2ð Þ� *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup>* <sup>2</sup> ,

<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup>* � �*:*

*D t*ð Þþ , *x t*ð Þ, *y t*ð Þ *E t*ð Þ , *x t*ð Þ, *y t*ð Þ " #*dt*<sup>þ</sup>

3 7 5 *d*wð Þ*t* , (38)

(40)

(42)

�*η*1*N*2ð Þ*<sup>t</sup>* , (39)

�*η*1*x*2ð Þ*<sup>t</sup> :* (41)

*g*1ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*2ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

and genetic adaptation is inconsequential.

*Invariants for a Dynamical System with Strong Random Perturbations*

�

*N*�*ε*<sup>2</sup> <sup>1</sup> ð Þ*t e*

We can introduce the stochastic model as a form

*dx*1ðÞ¼ *t* ð Þ *ε*1*x*1ð Þ�*t η*1*x*1ð Þ*t x*2ðÞþ*t s*1ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t dt*þ

*dx*2ðÞ¼ � *t* ð Þ *ε*2*x*2ðÞþ*t η*2*x*1ð Þ*t x*2ðÞþ*t s*2ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t dt*þ

<sup>1</sup> ð Þ*t e*

*<sup>x</sup>*ð Þ¼ <sup>0</sup> *<sup>y</sup>*ð Þ¼ <sup>0</sup> 1. The function *u t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>x</sup>*�<sup>1</sup>*ex* � *<sup>y</sup>*<sup>2</sup>*e*�2*<sup>y</sup>* is a first integral, *h t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>y</sup>* � *<sup>x</sup>* <sup>þ</sup> *<sup>e</sup>*�*<sup>t</sup>* and *q t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>x</sup>* are complementary functions.

�*z*1ð Þ *<sup>t</sup>*,*x*,*y*,*<sup>γ</sup>* <sup>1</sup> � *<sup>z</sup>*�<sup>1</sup>

*dy t*ð Þ " # <sup>¼</sup> *A t*ð Þþ , *x t*ð Þ, *y t*ð Þ *B t*ð Þþ , *x t*ð Þ, *y t*ð Þ *C t*ð Þ , *x t*ð Þ, *y t*ð Þ

*x t*ð Þ*e*�*y t*ð Þ *<sup>y</sup>*<sup>2</sup> ð Þ ðÞ�*<sup>t</sup>* <sup>2</sup>*y t*ð Þ

*x t*ð Þ

*ex t*ð Þ *x t*ð Þ � *<sup>e</sup>*

Then, the constructed SDE system includes only Wiener perturbation:

We cannot find an analytical solution of the differential equations system

<sup>þ</sup>*b*1ð Þ *<sup>t</sup>*, *<sup>x</sup>*1ð Þ*<sup>t</sup>* , *<sup>x</sup>*2ð Þ*<sup>t</sup> <sup>d</sup>*w1ð Þþ*<sup>t</sup>* <sup>Ð</sup>

<sup>þ</sup>*b*2ð Þ *<sup>t</sup>*, *<sup>x</sup>*1ð Þ*<sup>t</sup>* , *<sup>x</sup>*2ð Þ*<sup>t</sup> <sup>d</sup>*w2ðÞþ*<sup>t</sup>* <sup>Ð</sup>

*u t*ð Þ¼ , **<sup>x</sup>**ð Þ*<sup>t</sup> <sup>x</sup>*�*ε*<sup>2</sup>

• Predators have limitless appetite.

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

Eq. (38) has the invariant function

*x*1ð Þ¼ 0 *N*1, *x*2ð Þ¼ 0 *N*2,

*<sup>∂</sup>z*1ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup>*

*<sup>∂</sup>z*2ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>γ</sup>*

*<sup>∂</sup><sup>γ</sup>* <sup>¼</sup> *<sup>e</sup>*

*<sup>∂</sup><sup>γ</sup>* ¼ �*<sup>e</sup>*

þ

2 6 4

the two species.

where *C* ¼ *const*.

with condition

8 >><

>>:

*dx t*ð Þ

**77**

8

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>>>>>>>>>>:

**Figure 2.** *Numerical solution for Eq.(36) with jumps.*

In such a way we could use the system of differential equations

$$\begin{cases} d\boldsymbol{y}\_1(t) = 2\boldsymbol{e}^{-t}dt, \\\\ d\boldsymbol{y}\_2(t) = \boldsymbol{y}\_1(t)d\mathbf{w}(t) + \int \boldsymbol{\eta}\mathbf{y}\_1(t)\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\\\ d\boldsymbol{y}\_3(t) = -2\boldsymbol{e}^{-t}dt + \boldsymbol{\eta}\_1(t)d\mathbf{w}(t) - \int \boldsymbol{\eta}\mathbf{y}\_1(t)\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\\\ \boldsymbol{y}(0) = \boldsymbol{y}\_0, \end{cases} \tag{37}$$

as initial step for construction of stochastic SIR-model. A good choice of complementary functions *v t*ð Þ , *x*, *y*, *z* and *h t*ð Þ , *x*, *y*, *z* allows us to obtain such coefficients that ensure that the solution f g *x t*ð Þ, *y t*ð Þ, *z t*ð Þ of the differential equations system satisfy some reasonable limitations.

#### **9.4 The predator–prey model**

The Lotka - Volterra equations or the predator–prey equations used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.

The Lotka - Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:


Let us note: *N*1ð Þ*t* is the number of prey, and *N*2ð Þ*t* is the number of some predator, *ε*1, *ε*2, *η*<sup>1</sup> and *η*<sup>2</sup> are positive real parameters describing the interaction of the two species.

The populations change through time according to the pair of equations:

$$\begin{cases} dN\_1(t) = N\_1(t)(\varepsilon\_1 - \eta\_1 N\_2(t))dt, \\ dN\_2(t) = -N\_2(t)(\varepsilon\_2 - \eta\_2 N\_1(t))dt. \end{cases} \tag{38}$$

Eq. (38) has the invariant function

$$N\_1^{-\varepsilon\_1}(t)e^{\eta\_2 N\_1(t)} = \text{CN}\_2^{\varepsilon\_1}(t)e^{-\eta\_1 N\_2(t)},\tag{39}$$

where *C* ¼ *const*.

In such a way we could use the system of differential equations

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

*γy*1ð Þ*t ν*ð Þ *dt*, *dγ* ,

*γy*1ð Þ*t ν*ð Þ *dt*, *dγ* ,

(37)

*dt* <sup>þ</sup> *<sup>y</sup>*1ð Þ*<sup>t</sup> <sup>d</sup>*wðÞ�*<sup>t</sup>* <sup>Ð</sup>

as initial step for construction of stochastic SIR-model. A good choice of complementary functions *v t*ð Þ , *x*, *y*, *z* and *h t*ð Þ , *x*, *y*, *z* allows us to obtain such coefficients that ensure that the solution f g *x t*ð Þ, *y t*ð Þ, *z t*ð Þ of the differential equations system

The Lotka - Volterra equations or the predator–prey equations used to describe the dynamics of biological systems in which two species interact, one as a predator

The Lotka - Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey

• The food supply of the predator population depends entirely on the size of the

*dt*,

*dy*2ðÞ¼ *<sup>t</sup> <sup>y</sup>*1ð Þ*<sup>t</sup> <sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

• The prey population finds ample food at all times.

• The rate of change of population is proportional to its size.

*dy*1ðÞ¼ *<sup>t</sup>* <sup>2</sup>*e*�*<sup>t</sup>*

8 >>>>>>><

*Numerical solution for Eq.(36) with jumps.*

**Figure 2.**

>>>>>>>:

satisfy some reasonable limitations.

**9.4 The predator–prey model**

and the other as prey.

prey population.

populations:

**76**

*dy*3ðÞ¼� *<sup>t</sup>* <sup>2</sup>*e*�*<sup>t</sup>*

*y*ð Þ¼ 0 *y*0,

We can introduce the stochastic model as a form

$$\begin{cases} d\mathbf{x}\_{1}(t) = (\mathbf{e}\_{1}\mathbf{x}\_{1}(t) - \eta\_{1}\mathbf{x}\_{1}(t)\mathbf{x}\_{2}(t) + s\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t)))dt + \\ \qquad \quad + b\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t))d\mathbf{w}\_{1}(t) + \int\_{\mathbb{R}} \mathbf{g}\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t),\boldsymbol{\chi})\boldsymbol{\nu}(dt,d\boldsymbol{\chi}), \\\\ d\mathbf{x}\_{2}(t) = (-\varepsilon\_{2}\mathbf{x}\_{2}(t) + \eta\_{2}\mathbf{x}\_{1}(t)\mathbf{x}\_{2}(t) + s\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t)))dt + \\ \qquad \quad + b\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t))d\mathbf{w}\_{2}(t) + \int\_{\mathbb{R}} \mathbf{g}\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t),\boldsymbol{\chi})\boldsymbol{\nu}(dt,d\boldsymbol{\chi}), \\\\ \mathbf{x}\_{1}(0) = N\_{1}, \qquad \quad \mathbf{x}\_{2}(0) = N\_{2}, \end{cases} (40)$$

with condition

$$u(t, \mathbf{x}(t)) = \mathfrak{x}\_1^{-\varepsilon\_2}(t)e^{\eta\_2 \mathbf{x}\_1(t)} - \mathbf{C} \mathfrak{x}\_2^{\varepsilon\_1}(t)e^{-\eta\_1 \mathbf{x}\_2(t)}.\tag{41}$$

Let us assume that *ε*<sup>1</sup> ¼ 2, *ε*<sup>2</sup> ¼ 1, *η*<sup>1</sup> ¼ *η*<sup>2</sup> ¼ 1, and *C* ¼ 1, and initial condition is *<sup>x</sup>*ð Þ¼ <sup>0</sup> *<sup>y</sup>*ð Þ¼ <sup>0</sup> 1. The function *u t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>x</sup>*�<sup>1</sup>*ex* � *<sup>y</sup>*<sup>2</sup>*e*�2*<sup>y</sup>* is a first integral, *h t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>y</sup>* � *<sup>x</sup>* <sup>þ</sup> *<sup>e</sup>*�*<sup>t</sup>* and *q t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup> <sup>x</sup>* are complementary functions.

We cannot find an analytical solution of the differential equations system

$$\begin{cases} \frac{\partial \mathbf{z}\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}} = e^{-x\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})} \mathbf{z}\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) \mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) (\mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) - \mathbf{2}),\\ \frac{\partial \mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}} = -e^{-x\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})} \left(1 - \mathbf{z}\_1^{-1}(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})\right). \end{cases}$$

Then, the constructed SDE system includes only Wiener perturbation:

$$
\begin{aligned}
\begin{bmatrix} dx(t) \\ dy(t) \end{bmatrix} &= \begin{bmatrix} A(t, \mathbf{x}(t), y(t)) + B(t, \mathbf{x}(t), y(t)) + C(t, \mathbf{x}(t), y(t)) \\ & D(t, \mathbf{x}(t), y(t)) + E(t, \mathbf{x}(t), y(t)) \end{bmatrix} dt + \\ &+ \begin{bmatrix} \mathbf{x}(t)e^{-\mathbf{y}(t)}(y^2(t) - 2\mathbf{y}(t)) \\ \mathbf{e}^{\mathbf{x}(t)} - \mathbf{e}^{\mathbf{x}(t)} \end{bmatrix} d\mathbf{w}(t), \end{aligned} \tag{42}
$$

where

*A t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ 0*:*5*e* �*y t*ð Þ*x t*ð Þ *<sup>y</sup>*<sup>2</sup> ðÞ�*t* 2*y t*ð Þ*e* �*y t*ð Þ <sup>2</sup> , *B t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ 0*:*5*e* �*y t*ð Þ *x t*ð Þ*<sup>e</sup> x t*ð Þ � *<sup>e</sup> x t*ð Þ*x*�<sup>1</sup> ð Þ*t* <sup>2</sup> � <sup>4</sup>*y t*ð Þþ *<sup>y</sup>*<sup>2</sup> ð Þ*<sup>t</sup>* , *C t*ð Þ¼� , *x t*ð Þ, *y t*ð Þ *<sup>e</sup>*�*<sup>t</sup> <sup>e</sup>*�*y t*ð Þ *<sup>y</sup>*<sup>2</sup> ð Þ ð Þ�*<sup>t</sup>* <sup>2</sup>*y t*ð Þ *ex t*ð Þ*x*�1ðÞ�*<sup>t</sup>* <sup>2</sup>*y t*ð Þ*e*�*y t*ð Þ � *ex t*ð Þ*x*�2ðÞþ*<sup>t</sup> <sup>y</sup>*2ð Þ*<sup>t</sup> <sup>e</sup>*�*y t*ð Þ , *D t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ *<sup>e</sup>*�*<sup>t</sup> <sup>e</sup>x t*ð Þ *<sup>x</sup>*�1ðÞ�*<sup>t</sup> <sup>x</sup>*�<sup>2</sup> ð Þ ð Þ*<sup>t</sup> ex t*ð Þ*x*�1ðÞ�*<sup>t</sup>* <sup>2</sup>*y t*ð Þ*e*�*y t*ð Þ � *ex t*ð Þ*x*�2ðÞþ*<sup>t</sup> <sup>y</sup>*2ð Þ*<sup>t</sup> <sup>e</sup>*�*y t*ð Þ , *E t*ð Þ¼� , *x t*ð Þ, *y t*ð Þ 0*:*5*e x t*ð Þ*e* �*y t*ð Þ *y*<sup>2</sup> ðÞ�*<sup>t</sup>* <sup>2</sup>*y t*ð Þ <sup>1</sup> � *<sup>x</sup>*�<sup>1</sup> ð Þþ*<sup>t</sup> x t*ðÞ� <sup>2</sup> <sup>þ</sup> <sup>2</sup>*x*�<sup>1</sup> ð Þ*<sup>t</sup> :* (43)

Finally, we have the stochastic Lotka Volterra model associated to (38)

*<sup>N</sup>*1ð Þ*<sup>t</sup> <sup>e</sup>*�*N*2ð Þ*<sup>t</sup> <sup>N</sup>*<sup>2</sup>

*Invariants for a Dynamical System with Strong Random Perturbations*

*eN*1ð Þ*<sup>t</sup> <sup>N</sup>*1ð Þ*<sup>t</sup>* � *<sup>e</sup>*

<sup>¼</sup> *A t*ð Þþ , *N t*ð Þ *B t*ð Þþ , *N t*ð Þ *C t*ð Þ , *N t*ð Þ *D t*ð Þþ , *N t*ð Þ *E t*ð Þ , *N t*ð Þ " #*dt*<sup>þ</sup>

<sup>2</sup>ðÞ�*<sup>t</sup>* <sup>2</sup>*N*2ð Þ*<sup>t</sup>* � �

3 7 5 *d*wð Þ*t* , (44)

*N*1ð Þ*t*

where *A t*ð Þ , *N t*ð Þ , *B t*ð Þ , *N t*ð Þ , *C t*ð Þ , *N t*ð Þ , *D t*ð Þ , *N t*ð Þ , *E t*ð Þ , *N t*ð Þ are determined by

**Figures 3** and **4** show two realizations for numerical solution of Eq. (44). Another examples of a differential equation system construction and models see

The invariant method widens horizons for constructing and researching into mathematical models of real systems with the invariants that hold out under any

(*N t*ðÞ¼ ð Þ *N*1ð Þ*t* , *N*2ð Þ*t* ):

Eq.(43).

in [25–29].

**10. Conclusion**

**Author details**

**79**

Elena Karachanskaya

Far-Eastern State Transport University, Khabarovsk, Russia

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: elena\_chal@mail.ru

provided the original work is properly cited.

strong random disturbances.

*dN*1ð Þ*t dN*2ð Þ*t*

þ

2 6 4

" #

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

**Figure 3.** *Numerical simulation 1 for solution of Eq.(44).*

**Figure 4.** *Numerical simulation 2 for solution of Eq.(44).*

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

Finally, we have the stochastic Lotka Volterra model associated to (38) (*N t*ðÞ¼ ð Þ *N*1ð Þ*t* , *N*2ð Þ*t* ):

$$\begin{aligned} \begin{bmatrix} dN\_1(t) \\ dN\_2(t) \end{bmatrix} &= \begin{bmatrix} A(t, N(t)) + B(t, N(t)) + C(t, N(t)) \\ D(t, N(t)) + E(t, N(t)) \end{bmatrix} dt + \\ &+ \begin{bmatrix} N\_1(t)e^{-N\_2(t)} \left(N\_2^2(t) - 2N\_2(t)\right) \\ e^{N\_1(t)} - e^{N\_1(t)} \end{bmatrix} d\mathbf{w}(t), \end{aligned} \tag{44}$$

where *A t*ð Þ , *N t*ð Þ , *B t*ð Þ , *N t*ð Þ , *C t*ð Þ , *N t*ð Þ , *D t*ð Þ , *N t*ð Þ , *E t*ð Þ , *N t*ð Þ are determined by Eq.(43).

**Figures 3** and **4** show two realizations for numerical solution of Eq. (44).

Another examples of a differential equation system construction and models see in [25–29].
