The Jungle Universe: coupled cosmological models in a Lotka–Volterra framework

Abstract

In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaître universes is a special case of generalized Lotka–Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka–Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann–Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka–Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.

Appendices

Appendix 1: The Lotka–Volterra equation

The Lotka–Volterra equation is defined by a set of two coupled ordinary differential equation of the form

$$\begin{aligned} \left\{ \begin{array}{c} x_{1}^{\prime }=x_{1}\left( +r_{1}-a_{12}x_{2}\right) \\ x_{2}^{\prime }=x_{2}\left( -r_{2}+a_{21}x_{1}\right) \end{array} \right. \end{aligned}$$

where $x_{1}\left( t\right)$ and $x_{2}\left( t\right)$ are two function of the variable $t$, a prime indicate the $t$-derivative and $r_{1},\,r_{2},\,a_{12}$ and $a_{21}$ are four positive constants. It is one of the simplest model to describe interactive dynamical population between some prey (which number is $x_{1}$) and its predator (which number is $x_{2}$). The constant have intuitive interpretation:

• $r_{1}$ (resp. $r_{2}$) is the growth rate for preys (resp. predators) to increase (resp. decrease) when there is no predators (resp. preys);

• $a_{12}$ and $a_{21}$ are the coupling factors between the two populations; in ecological systems they are related to the mobility, the agressivity and other natural properties of the different concerned populations.

The properties of such a set of equation are very well known and a lot of classical books (see [24] and reference within) explain and demonstrate that starting with strictly positive initial conditions, the solution $x_{1}\left( t\right)$ and $x_{2}\left( t\right)$ are periodic functions. The orbits in the phase space are concentric closed curves centered on the equilibrium $\left( \tilde{x}_{1},\tilde{x}_{2}\right) =\left( \frac{r_{1}}{a_{12}},\frac{ r_{2}}{a_{21}}\right)$. These curves correspond to the isocontours of the Ljapunov function $V\left( x_{1},x_{2}\right) =a_{21}x+a_{12}y- \left( r_{2}\ln x+r_{1}\ln y\right)$.

When $r_{1},\,r_{2},\,a_{12}$ and $a_{21}$ have different signs the situation is changed and the system can describe other situations like competition, symbiose, etc (see [24]).

Lotka–Volterra systems is often considered as an unrealistic model in the context of population dynamics because there is no limitation when one specie is ruled out. A better model could be implemented introducing a logistic limitation such that

$$\begin{aligned} \left\{ \begin{array}{c} x_{1}^{\prime }=x_{1}\left( +r_{1}-a_{11}x_{1}-a_{12}x_{2}\right) \\ x_{2}^{\prime }=x_{2}\left( -r_{2}+a_{21}x_{1}-a_{22}x_{2}\right) \end{array} \right. \end{aligned}$$

The sign of the new parameters $a_{11}$ and $a_{22}$ is always related to the physical nature of the dynamical system. When all constants are positives, the system admits a general treatment: the Lotka–Volterra curves are generally replaced by spirals which converge to the equilibrium $\left( x^{*}\ne 0,y^{*}\ne 0\right)$ solution of the system

$$\begin{aligned} \left\{ \begin{array}{l} r_{1}=+a_{11}x_{1}+a_{12}x_{2} \\ r_{2}=-a_{21}x_{1}+a_{22}x_{2} \end{array} \right. \end{aligned}$$

When the matrix $A=\left( a_{ij}\right)$ is non invertible $\left( \text { singular}\right) \$the situation is called degenerated and equilibria lie on the axes.

Such models could be generalized to $n$-dimensional systems introducing the generalized Lotka–Volterra equation occuring in this paper. Less things can be said about dynamical properties of such systems when the coefficient of $\mathbf {r}$ and $A$ are generic ones, in particular the dynamics can change strongly giving rise to cyclic competition or cooperation [33].

Appendix 2: Counter example: a linear center which is actually nonlinearly unstable

Consider the dynamical system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x}=-y+x^{3}\\ \dot{y}=x+y^{3} \end{array} \right. \end{aligned}$$

The origin is an equilibrium, the eigenvalues of the jacobian near this equilibrium are $\pm$ i. In the linear approximation the origin seems to be a center. But as the eigenvalues have no real part, the system is not hyperbolic and almost nothing can be assumed about the non linear dynamics considering only the linear one around the equilibrium. In this particular case one can prove that the origin is a repulsive focus, and it is then actually unstable. As a matter of fact, considering the intersection $M$ between an orbit and the circle $x^{2}+y^{2}=R^{2}$. The angle $\alpha$ between the tangent in $M$ to the orbit and the tangent in $M$ to the circle is given for any radius $R$ by the expression

$$\begin{aligned} \cos \alpha =\left( \dot{x},\dot{y}\right) \cdot \left( 2x,2y\right) =\left( -y+x^{3},x+y^{3}\right) \cdot \left( 2x,2y\right) =2x^{4}+2y^{4}>0 \end{aligned}$$

Hence, each orbit is going out from any circle of radius $R>0$ and the origin is unstable.

Appendix 3: Proof of the stability of the $\Omega _{\mathrm{d}}-\Omega _{\mathrm{e}}$ plane

The aim of this section is to provide a simple proof of the attractiveness of the $\Omega _{\mathrm{d}}-\Omega _{\mathrm{e}}$ plane for all orbits whose initial conditions belong to the hyper-tetrahedron:

$$\begin{aligned} T_{4}=\left\{ \Omega _{\mathrm{d}}>0\right\} \cup \left\{ \Omega _{\mathrm{e}}>0\right\} \cup \left\{ \Omega _{\mathrm{r}}>0\right\} \cup \left\{ \Omega _{\mathrm{b}}>0\right\} \cup \left\{ \Omega _{\mathrm{d}}+\Omega _{\mathrm{e}}+\Omega _{\mathrm{r}}+\Omega _{\mathrm{b} }<1\right\} . \end{aligned}$$

Let us recall the ODE system describing the equations of motion:

$$\begin{aligned} \left\{ \begin{array}{l} \Omega _{\mathrm{d}}^{\prime }=\Omega _{\mathrm{d}}\left[ \Omega _{\mathrm{d} }+(\varepsilon +1+3\omega _{\mathrm{e}})\Omega _{\mathrm{e}}+2\Omega _{\mathrm{r} }+\Omega _{\mathrm{b}}-1\right] \\ \Omega _{\mathrm{e}}^{\prime }=\Omega _{\mathrm{e}}\left[ (1-\varepsilon )\Omega _{\mathrm{d}}+\left( 1+3\omega _{\mathrm{e}}\right) \Omega _{\mathrm{e}}+2\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}-1-3\omega _{\mathrm{e} }\right] \\ \Omega _{\mathrm{r}}^{\prime }=\Omega _{\mathrm{r}}\left[ \Omega _{\mathrm{d} }+\left( 1+3\omega _{\mathrm{e}}\right) \Omega _{\mathrm{e}}+2\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}-2\right] \\ \Omega _{\mathrm{b}}^{\prime }=\Omega _{\mathrm{b}}\left[ \Omega _{\mathrm{d} }+\left( 1+3\omega _{\mathrm{e}}\right) \Omega _{\mathrm{e}}+2\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}-1\right] \end{array} \right. \end{aligned}$$

(21)

To prove our claim we need to prove first the invariance with respect to the flow (21) of the hyper-tetrahedron $T_{4}$. The invariance of each coordinates hyperplanes is trivial and follows straightforwardly from (21). For instance any solution such that $\Omega _{\mathrm{d} }(0)=0$ will have $\Omega _{\mathrm{d}}(\lambda )=0$ for all $\lambda$, then using the uniqueness of the Cauchy problem we can ensure that any solution with $\Omega _{\mathrm{d}}(0)>0$ will never cross the hyperplane $\Omega _{\mathrm{d}}=0$. A very similar analysis can be performed for the remaining cases.

Let us now consider the remaining piece of the boundary of $T_{4}$, that is the hyperplane$\{\Omega _{\mathrm{d}}+\Omega _{\mathrm{e}}+\Omega _{\mathrm{r} }+\Omega _{\mathrm{b}}=1\}$. A straightforward computation gives:

$$\begin{aligned} \left( \Omega _{\mathrm{d}}\!+\!\Omega _{\mathrm{e}}\!+\!\Omega _{\mathrm{r}} \!+\!\Omega _{\mathrm{b}}\right) ^{\prime }=\left[ \Omega _{\mathrm{d}} \!+\!(1\!+\!3\omega _{\mathrm{e}})\Omega _{\mathrm{e}}\!+\!2\Omega _{\mathrm{r}} \!+\!\Omega _{\mathrm{b}}\right] \left[ \Omega _{\mathrm{d}}\!+\!\Omega _{\mathrm{e} }\!+\!\Omega _{\mathrm{r}}\!+\!\Omega _{\mathrm{b}}-1\right] , \end{aligned}$$

thus any solution with initial conditions

$$\begin{aligned} \Omega _{\mathrm{d}}(0)+\Omega _{\mathrm{e}}(0)+\Omega _{\mathrm{r}} (0)+\Omega _{\mathrm{b}}(0)=1, \end{aligned}$$

will always satisfies the constraint

$$\begin{aligned} \Omega _{\mathrm{d}}(\lambda )+\Omega _{\mathrm{e}}(\lambda )+\Omega _{\mathrm{r} }(\lambda )+\Omega _{\mathrm{b}}(\lambda )=1\quad \forall \lambda . \end{aligned}$$

Thus once again the uniqueness result of the Cauchy problem implies that any solution such that $\Omega _{\mathrm{d}}(0)+\Omega _{\mathrm{e}}(0)+\Omega _{\mathrm{r}}(0)+\Omega _{\mathrm{b}}(0)<1$, will never reach the hyperplane $\Omega _{\mathrm{d}}+\Omega _{\mathrm{e}}+\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}=1$.

Finally putting together the above partial results, we can conclude that any orbit with initial condition inside $T_{4}$ will never leave it.

A by-product of the invariance of the tetrahedron is that orbits inside $T_{4}$ will always have positive projections on the axes. This allows us to compute the distance from the plane $\left( \Omega _{\mathrm{d}},\Omega _{\mathrm{e}}\right)$ using the linear function $F(\Omega _{\mathrm{r}},\Omega _{\mathrm{b}})=\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}$, which is zero if and only if $\Omega _{\mathrm{r}}=\Omega _{\mathrm{b}}=0$, that is the point belongs to the plane $\left( \Omega _{\mathrm{d}},\Omega _{\mathrm{e}}\right)$.

We can then compute the Lie derivative of $F$ and prove that its restriction to $T_{4}$ is strictly negative, hence $F(\Omega _{\mathrm{r}}(\lambda ),\Omega _{\mathrm{b}}(\lambda ))\rightarrow 0$ for $\lambda \rightarrow +\infty$ and because of the positiveness of $\Omega _{\mathrm{r}}\left( \lambda \right)$ and $\Omega _{\mathrm{b}}\left( \lambda \right)$ we can conclude that both $\Omega _{\mathrm{r}}\left( \lambda \right)$ and $\Omega _{\mathrm{b}}\left( \lambda \right)$ goes asymptotically to zero.

To prove the latter claim let us compute the derivative of $F$ along the flow of (21):

$$\begin{aligned} \left. \frac{dF}{dt}\right| _{\mathrm{flow}}=\left[ \Omega _{\mathrm{d} }+(1+3\omega _{\mathrm{e}})\Omega _{\mathrm{e}}+2\Omega _{\mathrm{r}} +\Omega _{\mathrm{b}}-1\right] \left[ \Omega _{\mathrm{r}}+\Omega _{\mathrm{b} }\right] -\Omega _{\mathrm{r}}, \end{aligned}$$

because of our previous result $\Omega _{\mathrm{d}}(\lambda )+\Omega _{\mathrm{b}}(\lambda )-1<-\Omega _{\mathrm{e}}(\lambda )-\Omega _{\mathrm{r} }(\lambda )$ for all $\lambda$. Using this inequality we get

$$\begin{aligned} \left. \frac{dF}{dt}\right| _{\mathrm{flow}}<\left( 3\omega _{\mathrm{e} }\Omega _{\mathrm{e}}\!+\!\Omega _{\mathrm{r}}\right) \left( \Omega _{\mathrm{r} }\!+\!\Omega _{\mathrm{b}}\right) -\Omega _{\mathrm{r}}\,=3\omega _{\mathrm{e} }\Omega _{\mathrm{e}}\left( \Omega _{\mathrm{r}}\!+\!\Omega _{\mathrm{b}}\right) \!+\!\left( \Omega _{\mathrm{r}}\!+\!\Omega _{\mathrm{b}}-1\,\right) \Omega _{\mathrm{r}} \end{aligned}$$

let us observe that the right hand side is strictly negative, in fact

$$\begin{aligned} 3\omega _{\mathrm{e}}\Omega _{\mathrm{e}}(\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}})<0 \end{aligned}$$

provided and $\Omega _{\mathrm{e}}$ is associated to the dark energy $\left( \omega _{\mathrm{e}}<-\frac{1}{3}<0\right)$ and

$$\begin{aligned} \Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}-1<\Omega _{\mathrm{d}}+\Omega _{\mathrm{e}}+\Omega _{\mathrm{r}}+\Omega _{\mathrm{b}}-1<0. \end{aligned}$$

This concludes our proof of the attractiveness of the dark plane for all orbits whose initial conditions belong to $T_{4}$.