# Integrable Deformations and Dynamical Properties of Systems with Constant Population

## Abstract

**:**

## 1. Introduction

## 2. Hamilton–Poisson Formulations and Integrable Deformations

**Proposition**

**1.**

- (i)
- System$$\dot{x}={C}_{z}-{C}_{y},\phantom{\rule{3.33333pt}{0ex}}\dot{y}={C}_{x}-{C}_{z},\phantom{\rule{3.33333pt}{0ex}}\dot{z}={C}_{y}-{C}_{x},$$
- (ii)
- C is also a constant of motion of system (2).
- (iii)
- $(P,{\Pi}_{C},H)$ is a Hamilton–Poisson formulation of system (2), where the Poisson structure ${\Pi}_{C}$ is given by$${\Pi}_{C}=\left(\right)open="["\; close="]">\begin{array}{ccc}0& {C}_{z}& -{C}_{y}\\ -{C}_{z}& 0& {C}_{x}\\ {C}_{y}& -{C}_{x}& 0\end{array}$$

**Proposition**

**2.**

**Proposition**

**3.**

- (i)
- (ii)
- If H and $C+g\beta $ are functionally independent on P, then a family of integrable deformations of system (1) is given by:$$\dot{x}={f}_{1}+g\nu ({\beta}_{z}-{\beta}_{y}),\phantom{\rule{3.33333pt}{0ex}}\dot{y}={f}_{2}+g\nu ({\beta}_{x}-{\beta}_{z}),\phantom{\rule{3.33333pt}{0ex}}\dot{z}={f}_{3}+g\nu ({\beta}_{y}-{\beta}_{x}),$$
- (iii)

**Proof.**

## 3. A Particular Case: Polynomial Kolmogorov Systems

**Proposition**

**4.**

**Proof.**

**Remark**

**1.**

**Proposition**

**5.**

- (i)
- The three-dimensional Lotka–Volterra system with constant population (8) has the Hamilton–Poisson formulation $(P,\Pi ,H),$ where $P={(0,\infty )}^{3},$$H(x,y,z)=x+y+z,$ and the Poisson structure$$\Pi =\left(\right)open="["\; close="]">\begin{array}{ccc}0& cxy& -bxz\\ -cxy& 0& ayz\\ bxz& -ayz& 0\end{array}$$
- (ii)
- If $\beta \in {C}^{1}\left(P\right)$ such that H and $C+g\beta $ are functionally independent on P, then a family of integrable deformations of Lotka–Volterra system (8) is given by$$\left(\right)open="\{"\; close>\begin{array}{c}\dot{x}=x(cy-bz)+g{x}^{1-a}{y}^{1-b}{z}^{1-c}({\beta}_{z}-{\beta}_{y})\hfill \\ \dot{y}=y(-cx+az)+g{x}^{1-a}{y}^{1-b}{z}^{1-c}({\beta}_{x}-{\beta}_{z})\hfill \\ \dot{z}=z(bx-ay)+g{x}^{1-a}{y}^{1-b}{z}^{1-c}({\beta}_{y}-{\beta}_{x})\hfill \end{array}$$

**Proposition**

**6.**

**Proposition**

**7.**

**Proof.**

## 4. Dynamical Properties of the Three-Dimensional Lotka–Volterra System with Constant Population

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Proposition**

**8.**

**Proof.**

**Remark**

**3.**

**Proposition**

**9.**

**Proof.**

- (i)
- $dF(aM,bM,cM)=0;$
- (ii)
- ${d}^{2}{F(aM,bM,cM)|}_{W\times W}=-{a}^{a-1}{b}^{b-1}{c}^{c}{M}^{a+b+c-2}(b\phantom{\rule{0.166667em}{0ex}}d{x}^{2}+a\phantom{\rule{0.166667em}{0ex}}d{y}^{2})$, which is negative definite for all $M>0.$

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Proposition**

**12.**

**Proof.**

**Proposition**

**13.**

**Proof.**

**Remark**

**6.**

**Proposition**

**14.**

- (i)
- If $({h}_{1},{h}_{2})\in {\Sigma}_{1}^{s}$, then ${\mathcal{F}}_{({h}_{1},{h}_{2})}=\left\{(aM,bM,cM)\right\},$ where $M={\displaystyle \frac{{h}_{1}}{a+b+c}},$ that is, a stable equilibrium state. In addition, ${\mathcal{F}}_{(0,0)}=\left\{(0,0,0)\right\}.$
- (ii)
- If $({h}_{1},{h}_{2})\in {\Sigma}_{2}^{u}$, then ${\mathcal{F}}_{({h}_{1},{h}_{2})}$ is the triangle with vertices at ${E}_{M}^{2}(M,0,0),{E}_{M}^{3}(0,M,0),{E}_{M}^{4}(0,0,M),$ where $M={h}_{1},$ that is, three unstable equilibrium states and the cycle of heteroclinic orbits that connect them (see Proposition 13).
- (iii)
- If $({h}_{1},{h}_{2})\in {\Sigma}^{p}$, then ${\mathcal{F}}_{({h}_{1},{h}_{2})}=\left(\right)open="\{"\; close="\}">(x,y,z)\in {[0,\infty )}^{3}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}x+y+z={h}_{1},{x}^{a}{y}^{b}{z}^{c}={h}_{2}$ that is, a periodic orbit.

**Proof.**

**Remark**

**7.**

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**

**Left**: The intersection of level sets: a periodic orbit.

**Right**: A local foliation of the level set ${S}_{h}=\{(x,y,z)\in {\mathbb{R}}^{3}:x+y+z=h,x,y,z\ge 0\}$ by periodic orbits around the stable equilibrium $(\frac{ah}{a+b+c},\frac{bh}{a+b+c},\frac{ch}{a+b+c})$.

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Lăzureanu, C.
Integrable Deformations and Dynamical Properties of Systems with Constant Population. *Mathematics* **2021**, *9*, 1378.
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Lăzureanu C.
Integrable Deformations and Dynamical Properties of Systems with Constant Population. *Mathematics*. 2021; 9(12):1378.
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2021. "Integrable Deformations and Dynamical Properties of Systems with Constant Population" *Mathematics* 9, no. 12: 1378.
https://doi.org/10.3390/math9121378