📂Dynamics

Lotka-Volterra Competition Model

Overview

The Lotka-Volterra competition model is a model capable of explaining the Principle of Competitive Exclusion, particularly describing situations where two groups inhibit each other. For instance, it can be applied to the relationship between rabbits and sheep sharing the same pasture or to the massacres between two rival tribes.

Model¹

$$\begin{align*} \dot{x_{1}} =& r_{1} x_{1} {{ K_{1} - x_{1} - \beta_{12} x_{2} } \over { K_{1} }} \\ \dot{x_{2}} =& r_{2} x_{2} {{ K_{2} - x_{2} - \beta_{21} x_{1} } \over { K_{2} }} \end{align*}$$

Variables

• $x_{1}(t)$: Represents the population of group $x_{1}$ at time $t$.
• $x_{2}(t)$: Represents the population of group $x_{2}$ at time $t$.

Parameters

• $r_{k}>0$: The intrinsic growth rate of $x_{k}$.
• $K_{k}>0$: The carrying capacity of the environment for $x_{k}$.
• $\beta_{ij} / K_{i} >0$: The Competition Coefficient for $x_{i}$.

Derivation

$$\dot{N} = {{ r } \over { K }} N ( K - N)$$

Let’s start from the logistic growth model. Assume initially that both populations $x_{k}$ grow at a reproduction rate of $r_{k}>0$ in the absence of predators and that their growth is limited by the carrying capacity $K_{k}$ of the environment. This is not truly a system of equations, but it can be expressed as follows:

$$\begin{align*} \dot{x_{1}} =& {{ r_{1} } \over { K_{1} }} x_{1} \left( K_{1} - x_{1} \right) \\ \dot{x_{2}} =& {{ r_{2} } \over { K_{2} }} x_{2} \left( K_{2} - x_{2} \right) \end{align*}$$

To this, we want to add terms representing the harm that each group causes to the other. The force hindering the growth of the opponents should be proportional to the strength of one’s own group. Therefore, for some constant $\beta > 0$, it can be expressed as follows:

$$\begin{align*} \dot{x_{1}} =& {{ r_{1} } \over { K_{1} }} x_{1} \left( K_{1} - x_{1} - \beta x_{2} \right) \\ \dot{x_{2}} =& {{ r_{2} } \over { K_{2} }} x_{2} \left( K_{2} - x_{2} - \beta x_{1} \right) \end{align*}$$

This means that the growth rate slows down due to checks from the rival group, even though the environmental capacity is not fully utilized. However, as the extent to which the two species affect each other may not be the same, specifying $\beta$ further completes the system as follows:

$$\begin{align*} \dot{x_{1}} =& {{ r_{1} } \over { K_{1} }} x_{1} \left( K_{1} - x_{1} - \beta_{12} x_{2} \right) \\ \dot{x_{2}} =& {{ r_{2} } \over { K_{2} }} x_{2} \left( K_{2} - x_{2} - \beta_{21} x_{1} \right) \end{align*}$$

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Equilibrium Points

• Deadlock $$\left( K_{1}, 0 \right) \\ \left( 0, K_{2} \right)$$

• Stalemate when $\beta_{12}\beta_{21} \ne 1$ $$\left( { { K_{1} - \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } }, { { K_{2} - \beta_{21} K_{1} } \over { 1 - \beta_{21} \beta_{12} } } \right)$$

In the Lotka-Volterra competition model, non-trivial equilibrium points consist of circumstances where a deadlock or stalemate occurs.

• A deadlock implies that in the competition between the two groups, one group completely eradicates the other, following the logistic growth model.
• A stalemate means that both groups are perfectly matched in strength, achieving a balance and coexistence. An interesting aspect in the formula is that the parameters intended during the modeling phase transparently affect the coordinates of the equilibrium points.
• However, these expressions are used for explanation in this post and not widely used.

Existence

$$\begin{align*} 0 =& {{ r_{1} } \over { K_{1} }} x_{1} \left( K_{1} - x_{1} - \beta_{12} x_{2} \right) \\ 0 =& {{ r_{2} } \over { K_{2} }} x_{2} \left( K_{2} - x_{2} - \beta_{21} x_{1} \right) \end{align*}$$

Let’s find the equilibrium points satisfying this.

Deadlock

Without loss of generality, assuming $x_{1} \ne 0, x_{2} = 0$,

$$\begin{align*} 0 =& {{ r_{1} } \over { K_{1} }} x_{1} \left( K_{1} - x_{1} \right) \\ 0 =& 0 \end{align*}$$

we simply obtain the equilibrium point $\left( K_{1}, 0 \right)$. Similarly, $\left( 0, K_{2} \right)$ can be found.

Stalemate

Assuming $x_{1} \ne 0, x_{2} \ne 0$ now, from the first equation we take $x_{1}$, from the second $x_{2}$, and then by eliminating the multiplied constants we simplify as follows:

$$\begin{align*} 0 =& K_{1} - x_{1} - \beta_{12} x_{2} \\ 0 =& K_{2} - x_{2} - \beta_{21} x_{1} \end{align*}$$

Finding solutions to this system of equations is similar to finding the intersection of two lines.

Moving the variables from the right side to the left side and expressing it in matrix form yields:

$$\begin{bmatrix} 1 & \beta_{12} \\ \beta_{21} & 1 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix} K_{1} \\ K_{2} \end{bmatrix}$$

If $\beta_{12}\beta_{21} \ne 1$, an inverse matrix exists, and

$$\begin{bmatrix} 1 & \beta_{12} \\ \beta_{21} & 1 \end{bmatrix}^{-1} = { { 1 } \over { 1 - \beta_{12} \beta_{21} } } \begin{bmatrix} 1 & - \beta_{12} \\ - \beta_{21} & 1 \end{bmatrix}$$

hence,

$$\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = { { 1 } \over { 1 - \beta_{12} \beta_{21} } } \begin{bmatrix} 1 & - \beta_{12} \\ - \beta_{21} & 1 \end{bmatrix} \begin{bmatrix} K_{1} \\ K_{2} \end{bmatrix}$$

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Stability

$${ { \partial } \over { \partial x_{1} } } \left[ r_{1} x_{1} {{ K_{1} - x_{1} - \beta_{12} x_{2} } \over { K_{1} }} \right] = { { r_{1} } \over { K_{1} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) \\ { { \partial } \over { \partial x_{2} } } \left[ r_{1} x_{1} {{ K_{1} - x_{1} - \beta_{12} x_{2} } \over { K_{1} }} \right] = - { { r_{1} } \over { K_{1} } } \beta_{12} x_{1} \\ { { \partial } \over { \partial x_{1} } } \left[ r_{2} x_{2} {{ K_{2} - x_{2} - \beta_{21} x_{1} } \over { K_{2} }} \right] = - { { r_{2} } \over { K_{2} } } \beta_{21} x_{2} \\ { { \partial } \over { \partial x_{2} } } \left[ r_{2} x_{2} {{ K_{2} - x_{2} - \beta_{21} x_{1} } \over { K_{2} }} \right] = { { r_{2} } \over { K_{2} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right)$$

Thus, the Jacobian is as follows:

$$J = \begin{bmatrix} { { r_{1} } \over { K_{1} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) & - { { r_{1} } \over { K_{1} } } \beta_{12} x_{1} \\ - { { r_{2} } \over { K_{2} } } \beta_{21} x_{2} & { { r_{2} } \over { K_{2} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) \end{bmatrix}$$

Deadlock

Without loss of generality, let’s consider only the case where $\left( K_{1} , 0 \right)$.

$$\begin{bmatrix} { { r_{1} } \over { K_{1} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) & - { { r_{1} } \over { K_{1} } } \beta_{12} x_{1} \\ - { { r_{2} } \over { K_{2} } } \beta_{21} x_{2} & { { r_{2} } \over { K_{2} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) \end{bmatrix}_{\left( K_{1} , 0 \right)} = \begin{bmatrix} - r_{1} & - \beta_{12} r_{1} \\ 0 & - \beta_{21} r_{2} \end{bmatrix}$$

The eigenvalues are solutions of

$$\begin{align*} \det \left( J - \lambda I \right) &= \left( - r_{1} - \lambda \right) \left( - \beta_{21} r_{2} - \lambda \right) + r_{1} \beta_{12} \\ =& \lambda^{2} + \left( r_{1} + \beta_{21} r_{2} \right) + r_{1} r_{2} \beta_{21} + r_{1} \beta_{12} \\ =& 0 \end{align*}$$

According to the quadratic formula, given that $r_{1}$, $r_{2}$, $\beta_{12}$, $\beta_{21}$ are all positive, it’s impossible for the real part to be positive as the imaginary part or real root, if present, cannot compensate enough for $- \left( r_{1} + \beta_{21} r_{2} \right)$. Therefore, the real parts must be negative, making equilibrium points $\left( K_{1} , 0 \right)$ and $\left( 0, K_{2} \right)$ stable.

Stalemate

If $\left( x_{1}, x_{2} \right) = \left( { { K_{1} - \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } }, { { K_{2} - \beta_{21} K_{1} } \over { 1 - \beta_{21} \beta_{12} } } \right)$ then

$$\begin{align*} K_{1} - 2 x_{1} - \beta_{12} x_{2} =& { { K_{1} - \beta_{12} \beta_{21} K_{1} - 2 K_{1} + 2 \beta_{12} K_{2} - \beta_{12} K_{2} + \beta_{12} \beta_{21} K_{1} } \over { 1 - \beta_{12} \beta_{21} } } \\ =& { { K_{1} - 2 K_{1} + 2 \beta_{12} K_{2} - \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } } \\ =& { { - K_{1} + \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } } \end{align*}$$

Similarly,

$$\begin{align*} K_{2} - 2 x_{1} - \beta_{21} x_{1} = { { - K_{2} + \beta_{21} K_{1} } \over { 1 - \beta_{12} \beta_{21} } } \end{align*}$$

The eigenvalues of the Jacobian will be solutions satisfying:

$$\begin{align*} 0 =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \det \left[ J_{\left( x_{1} , x_{2} \right)} - \lambda I \right] \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \left[ \left( { { r_{1} } \over { K_{1} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) - \lambda \right) \left( { { r_{2} } \over { K_{2} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) - \lambda \right) - { { r_{1} r_{2} } \over { K_{1} K_{2} } } \beta_{12} \beta_{21} x_{1} x_{2} \right] \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \left[ { { r_{1} } \over { K_{1} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} - { { K_{1} } \over { r_{1} } } \lambda \right) { { r_{2} } \over { K_{2} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} - { { K_{2} } \over { r_{2} } } \lambda \right) - { { r_{1} r_{2} } \over { K_{1} K_{2} } } \beta_{12} \beta_{21} x_{1} x_{2} \right] \\ =& \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} - { { K_{1} } \over { r_{1} } } \lambda \right) \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} - { { K_{2} } \over { r_{2} } } \lambda \right) - \beta_{12} \beta_{21} x_{1} x_{2} \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \lambda^{2} - \left[ { { K_{2} } \over { r_{2} } } \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) + { { K_{1} } \over { r_{1} } } \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) \right] \lambda \\ & + \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) - \beta_{12} \beta_{21} x_{1} x_{2} \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \lambda^{2} - \left[ { { K_{2} } \over { r_{2} } } { { - K_{1} + \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } } + { { K_{1} } \over { r_{1} } } { { - K_{2} + \beta_{21} K_{1} } \over { 1 - \beta_{12} \beta_{21} } } \right] \lambda \\ & + \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) - \beta_{12} \beta_{21} x_{1} x_{2} \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \lambda^{2} - { { 1 } \over { r_{1} r_{2} \left( 1 - \beta_{12} \beta_{21} \right) } } \left[ - r_{1} K_{1} K_{2} + r_{1} \beta_{12} K_{2}^{2} - r_{2} K_{2} K_{1} + r_{2} \beta_{21} K_{1}^{2} \right] \lambda \\ & + \left( K_{1} - 2 x_{1} - \beta_{12} x_{2} \right) \left( K_{2} - 2 x_{2} - \beta_{21} x_{1} \right) - \beta_{12} \beta_{21} x_{1} x_{2} \\ =& { { K_{1} K_{2} } \over { r_{1} r_{2} } } \lambda^{2} + { { 1 } \over { r_{1} r_{2} \left( 1 - \beta_{12} \beta_{21} \right) } } \left[ r_{1} K_{2} \left( K_{1} - \beta_{12} K_{2} \right) + r_{2} K_{1} \left( K_{2} - \beta_{21} K_{1} \right) \right] \lambda \\ & + { { - K_{1} + \beta_{12} K_{2} } \over { 1 - \beta_{12} \beta_{21} } } { { - K_{2} + \beta_{21} K_{1} } \over { 1 - \beta_{12} \beta_{21} } } - \beta_{12} \beta_{21} x_{1} x_{2} \end{align*}$$

As seen in the formula, the parameters that most affect the stability of the equilibrium points are $K_{1}$, $K_{2}$, $\beta_{12}$, $\beta_{21}$, depending on whether $K_{i}$ is greater or lesser than $\beta_{ij} K_{j}$. In an extreme case, if all $\beta_{ij}$ are $0$ such that there’s absolutely no competition, regardless of the opposing group, they will simply undergo individual logistic growth and converge to a stable equilibrium. However, if the competition is too severe, as soon as they diverge from the exact equilibrium, even slightly, the group with a slight advantage will try to annihilate the other. The term ‘instability’ fits perfectly in a mathematical expression. However, without knowing how the eigenvalues will be calculated based on the parameters, this equilibrium can be stable, unstable, or a saddle.

Example

$$\begin{align*} \dot{x_{1}} =& x_{1} \left( 3 - x_{1} - 2 x_{2} \right) \\ \dot{x_{2}} =& x_{2} \left( 2 - x_{2} - x_{1} \right) \end{align*}$$

The above system proposes the Lotka-Volterra competition model, considering $x_{1}$ as rabbits and $x_{2}$ as sheep. Although the two species don’t directly harm each other, it’s assumed they compete for food by sharing the same pasture. Since sheep are larger than rabbits, in terms of carrying capacity, rabbits are affected less by sheep than vice versa. The non-trivial solution of this system is as follows².

$$(3,0) \\ (0,2) \\ (1,1)$$

Especially for $(1,1)$, which becomes $J_{(1,1)}=\begin{bmatrix} -1 & -2 \\ -1 & -1 \end{bmatrix}$, the eigenvalues are $-1 \pm \sqrt{2}$, containing both positive and negative, making it a saddle.

Visual Understanding

The animation above depicts the vector field of the example. Regardless of the starting point, trajectories towards the equilibrium points can be observed, among which $(1,1)$ is a saddle, causing the points near it to head towards stable equilibrium points $(3,0)$ or $(0,2)$. This implies that without achieving perfect coexistence from the start, if one side is even slightly advantageous, the balance will gradually tilt in its favor.

Limitations

In this model, the case where the two populations achieve balance and coexist accounts for only a single point in the vast first quadrant. Despite competition, this is exceedingly extreme; indeed, this stems from the overly simplistic nature of the current Lotka-Volterra competition model. Naturally, using this model alone to approach real-world problems is impractical.