# Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model

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## Abstract

**:**

## 1. Introduction

## 2. Jacobi Last Multiplier and Its Properties

- (a)
- If one selects a different set of $n-1$ independent solutions ${\eta}_{1},\dots ,{\eta}_{n-1}$ of Equation (3), then the corresponding last multiplier N is linked to M by the relationship:$$N=M\frac{\partial ({\eta}_{1},\dots ,{\eta}_{n-1})}{\partial ({\omega}_{1},\dots ,{\omega}_{n-1})}.$$
- (b)
- Given a non-singular transformation of variables:$$\tau :\phantom{\rule{1.em}{0ex}}({x}_{1},{x}_{2},\dots ,{x}_{n})\u27f6({x}_{1}^{\prime},{x}_{2}^{\prime},\dots ,{x}_{n}^{\prime}),$$$${M}^{\prime}=M\frac{\partial ({x}_{1},{x}_{2},\dots ,{x}_{n})}{\partial ({x}_{1}^{\prime},{x}_{2}^{\prime},\dots ,{x}_{n}^{\prime})},$$
- (c)
- One can prove that each multiplier M is a solution of the following linear partial differential equation:$$\sum _{i=1}^{n}\frac{\partial \left(M{a}_{i}\right)}{\partial {x}_{i}}=0;$$
- (d)
- If one knows two Jacobi last multipliers ${M}_{1}$ and ${M}_{2}$ of Equation (3), then their ratio is a solution ω of (3) or, equivalently, a first integral of (4). Naturally, the ratio may be quite trivial, namely a constant; vice versa, the product of a multiplier ${M}_{1}$ times any solution ω yields another last multiplier ${M}_{2}={M}_{1}\omega $.

## 3. Lie Symmetries of System (1)–(2)

**Case (A)**

**Case (B)**

**Subcase (B.1)**

**Figure 1.**The amount of resources $R\equiv {w}_{1}$ and that of the population $P\equiv {w}_{2}$ for the values of the parameters $K=20000,\phantom{\rule{3.33333pt}{0ex}}c=0.01$, and for the initial conditions $P\left(0\right)=$ 1000, $R\left(0\right)=$ 20000.

**Figure 2.**The amount of resources $R\equiv {w}_{1}$ and that of the population $P\equiv {w}_{2}$ for the values of the parameters $K=40000,\phantom{\rule{3.33333pt}{0ex}}c=0.01$, and for the initial conditions $P\left(0\right)=$ 1000, $R\left(0\right)=$ 20000.

## 4. Jacobi Last Multipliers, Lagrangians and First Integrals of System (1)–(2)

**Remark 1:**It was shown in [7], Proposition 6, that if $c>a$, then the first integral (69) yields periodic orbits. In particular, if $a=1,c=2$, it is easy to show that the general solution depends on elliptic functions. ⧫

**Remark 2:**The case $a=2h,c=h$ is a particular example of Case (B), and consequently, Equation (24), i.e.,

**Remark 3:**The case $a=h=c$ is a particular example of Case (A), and consequently, Equation (24), i.e.,

**Remark 4:**The case $a=h=2c$ is another instance of Case (B), and consequently, Equation (24), i.e.,

**Case (A)**

**Case (B)**

**Subcase (B.1)**

## 5. Discussion and Final Remarks

- the first integral ${I}_{0}$ in (74) if $a=2h,c=h$;
- the first integral ${I}_{1a}$ in (84) and also the general solution (87) if $a=h=c$;
- the first integral ${I}_{1b}$ in (90) and also the general solution (94) if $a=h,h=2c$;
- the first integral ${I}_{A}$ in (98) if $h=\frac{a(2a-c)}{3a-2c}$, that corresponds to Proposition 3 ($h=2$, $c=4/3$);
- the first integral ${I}_{B}$ in (103) if $a=2c$, that corresponds to Proposition 1 ($h=3/4,c=1/2$).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Nucci, M.C.; Sanchini, G.
Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model. *Symmetry* **2015**, *7*, 1613-1632.
https://doi.org/10.3390/sym7031613

**AMA Style**

Nucci MC, Sanchini G.
Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model. *Symmetry*. 2015; 7(3):1613-1632.
https://doi.org/10.3390/sym7031613

**Chicago/Turabian Style**

Nucci, M.C., and G. Sanchini.
2015. "Symmetries, Lagrangians and Conservation Laws of an Easter Island Population Model" *Symmetry* 7, no. 3: 1613-1632.
https://doi.org/10.3390/sym7031613