# 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$${\Pi}_{C}=\left[\begin{array}{ccc}0& {C}_{z}& -{C}_{y}\\ -{C}_{z}& 0& {C}_{x}\\ {C}_{y}& -{C}_{x}& 0\end{array}\right],$$

**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[\begin{array}{ccc}0& cxy& -bxz\\ -cxy& 0& ayz\\ bxz& -ayz& 0\end{array}\right].$$
- (ii)
- If $\beta \in {C}^{1}\left(P\right)$ such that H and $C+g\beta $ are functionally independent on P, then a family of integrable deformations of Lotka–Volterra system (8) is given by$$\left\{\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}\right.,$$

**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 ${\mathcal{F}}_{({h}_{1},{h}_{2})}=\left\{(x,y,z)\in {[0,\infty )}^{3}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}x+y+z={h}_{1},{x}^{a}{y}^{b}{z}^{c}={h}_{2}\right\},$ 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

- Kermack, W.O.; McKendrick, A.G. A Contribution to the Mathematical Theory of Epidemics. Proc. R. Soc. A Math.
**1927**, 115, 700–721. [Google Scholar] - White, E.; Comiskey, C. Heroin epidemics, treatment and ODE modelling. Math. Biosci.
**2007**, 208, 312–324. [Google Scholar] [CrossRef] [PubMed] - Dauhoo, M.Z.; Korimboccus, B.S.N.; Issack, S.B. On the dynamics of illicit drug consumption in a given population. IMA J. Appl. Math.
**2013**, 78, 432–448. [Google Scholar] [CrossRef] - Bogoyavlenskij, O.I. Integrable Lotka—Volterra Systems. Regul. Chaotic Dyn.
**2008**, 13, 543–556. [Google Scholar] [CrossRef] - Kamp, P.H.V.; Kouloukas, T.E.; Quispel, G.R.W.; Tran, D.T.; Vanhaecke, P. Integrable and superintegrable systems associated with multi-sums of products. Proc. R. Soc. A
**2014**, 470, 20140481. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Galajinsky, A. Remark on integrable deformations of the Euler top. J. Math. Anal. Appl.
**2014**, 416, 995–997. [Google Scholar] [CrossRef] - Lăzureanu, C. On the Hamilton–Poisson realizations of the integrable deformations of the Maxwell-Bloch equations. C. R. Math.
**2017**, 355, 596–600. [Google Scholar] [CrossRef] - Lăzureanu, C. Hamilton–Poisson Realizations of the Integrable Deformations of the Rikitake System. Adv. Math. Phys.
**2017**, 2017, 4596951. [Google Scholar] [CrossRef] [Green Version] - Libermann, P.; Marle, C.-M. Symplectic Geometry and Analytical Mechanics; D. Reidel: Dordrecht, The Netherlands, 1987. [Google Scholar]
- Marsden, J.E.; Raţiu, T.S. Introduction to Mechanics and Symmetry, 2nd ed.; Texts in Applied Mathematics 17; Springer: New York, NY, USA, 1999. [Google Scholar]
- Puta, M. Hamiltonian Mechanical System and Geometric Quantization; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Tudoran, R.M. A normal form of completely integrable system. J. Geom. Phys.
**2012**, 62, 1167–1174. [Google Scholar] [CrossRef] - Gürses, M.; Guseinov, G.S.; Zheltukhin, K. Dynamical systems and Poisson structures. J. Math. Phys.
**2009**, 50, 112703. [Google Scholar] [CrossRef] [Green Version] - Lăzureanu, C. Integrable Deformations of Three-Dimensional Chaotic Systems. Int. J. Bifurc. Chaos
**2018**, 28, 1850066. [Google Scholar] [CrossRef] - Lotka, A.J. Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA
**1920**, 6, 410–415. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fečkan, M. A Generalization of Bendixson’s Criterion. Proc. Am. Math. Soc.
**2001**, 129, 3395–3399. [Google Scholar] [CrossRef] - Kolmogorov, A. Sulla teoria di Volterra della lotta per l’esistenza. G. Dell’ Ist. Ital. Degli Attuari
**1936**, 7, 74–80. [Google Scholar] - Sigmund, K. Kolmogorov and population dynamics. In Kolmogorov’s Heritage in Mathematics; Charpentier, É., Lesne, A., Nikolski, N.K., Eds.; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Bianca, C.; Pappalardo, F.; Pennisi, M.; Ragusa, M.A. Persistence Analysis in a Kolmogorov-type Model for Cancer-Immune System Competition. In Proceedings of the 11st International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2013), Rhodes, Greece, 21–27 September 2013; Volume 1558, pp. 1797–1800. [Google Scholar]
- Diz-Pita, E.; Llibre, J.; Otero-Espinar, M.V.; Valls, C. The zero-Hopf bifurcations in the Kolmogorov systems of degree 3 in R
^{3}. Commun. Nonlinear Sci. Numer. Simul.**2021**, 95, 105621. [Google Scholar] [CrossRef] - Grammaticos, B.; Moulin-Ollagnier, J.; Ramanic, A.; Strelcyn, J.-M.; Wojciechowski, S. Integrals of quadratic ordinary differential equations in R
^{3}: The Lotka–Volterra system. Phys. A Stat. Mech. Its Appl.**1990**, 163, 683–722. [Google Scholar] [CrossRef] - Labrunie, S. On the polynomial first integrals of the (a, b, c) Lotka–Volterra system. J. Math. Phys.
**1996**, 37, 5539–5550. [Google Scholar] [CrossRef] - Liang, X.; Jiang, J.F. The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka–Volterra systems. Nonlinearity
**2003**, 16, 785–801. [Google Scholar] [CrossRef] - Llibre, J.; Ramirez, R.; Ramirez, V. Integrability of a class of N-dimensional Lotka–Volterra and Kolmogorov systems. J. Differ. Equ.
**2020**, 269, 2503–2531. [Google Scholar] [CrossRef] - Llibre, J.; Valls, C. Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka–Volterra systems. Z. FÜr Angew. Math. Und Phys.
**2011**, 62, 761–777. [Google Scholar] [CrossRef] [Green Version] - Cairó, L.; Feix, M.R. Families of invariants of the motion for the Lotka–Volterra equations: The linear polynomials family. J. Math. Phys.
**1992**, 33, 2440. [Google Scholar] [CrossRef] - Tudoran, R.M.; Aron, A.; Nicoară, Ş. On a Hamiltonian Version of the Rikitake System. SIAM J. Appl. Dyn. Syst.
**2009**, 8, 454–479. [Google Scholar] [CrossRef] - Lotka, A.J. Analytical Theory of Biological Populations; The Plenum Series on Demographic Methods and Population Analysis; Plenum Press: New York, NY, USA, 1998. [Google Scholar]
- Diz-Pita, E.; Llibre, J.; Otero-Espinar, M.V. Phase portraits of a family of Kolmogorov systems depending on six parameters. Electron. J. Differ. Equ.
**2021**, 2021, 1–38. [Google Scholar] - Han, M.; Llibre, J.; Tian, Y. On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R
^{3}. Mathematics**2020**, 8, 1137. [Google Scholar] [CrossRef] - Llibre, J.; Martínez, Y.P. Dynamics of a family of Lotka–Volterra systems in R
^{3}. Nonlinear Anal.**2020**, 199, 111915. [Google Scholar] [CrossRef] - Bînzar, T.; Lăzureanu, C. On some dynamical and geometrical properties of the Maxwell-Bloch equations with a quadratic control. J. Geom. Phys.
**2013**, 70, 1–8. [Google Scholar] [CrossRef] - Bînzar, T.; Lăzureanu, C. A Rikitake type system with one control. Discret. Contin. Dyn. Syst. B
**2013**, 18, 1755–1776. [Google Scholar] [CrossRef] - Lăzureanu, C. The Real-Valued Maxwell–Bloch Equations with Controls: From a Hamilton–Poisson System to a Chaotic One. Int. J. Bifurc. Chaos
**2017**, 27, 1750143. [Google Scholar] [CrossRef] - Lăzureanu, C. On a Hamilton–Poisson Approach of the Maxwell-Bloch Equations with a Control, Mathematical Physics. Anal. Geom.
**2017**, 20, 20. [Google Scholar] - Lăzureanu, C.; Bînzar, T. A Rikitake type system with quadratic control. Int. J. Bifurc. Chaos
**2012**, 22, 1250274. [Google Scholar] [CrossRef] - Lăzureanu, C.; Bînzar, T. Some geometrical properties of the Maxwell-Bloch equations with a linear control. In Proceedings of the XIII-th International Conference on Mathematics and its Applications, Timişoara, Romania, 1–3 November 2012; pp. 151–158. [Google Scholar]
- Lăzureanu, C.; Bînzar, T. On a Hamiltonian version of controls dynamic for a drift-free left invariant control system on G
_{4}. Int. J. Geom. Methods Mod. Phys.**2012**, 9, 1250065. [Google Scholar] [CrossRef] - Arnold, V. Conditions for nonlinear stability of stationary plane curvilinear flows on an ideal fluid. Dokl. Akad. Nauk.
**1965**, 162, 773–777. [Google Scholar] - Llibre, J.; Zhang, X. Dynamics of Some Three-Dimensional Lotka—Volterra Systems. Mediterr. J. Math.
**2017**, 14, 126. [Google Scholar] [CrossRef] - Birtea, P.; Puta, M.; Tudoran, R.M. Periodic orbits in the case of a zero eigenvalue. Comptes Rendus Math.
**2007**, 344, 779–784. [Google Scholar] [CrossRef] [Green Version]

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