Problem 27 - Biocalculus Calculus for the Life Sciences

Question

Competition-colonization models In Exercise 7.Review.23 a metapopulation model for two species was introduced. The equations were $$\begin{aligned} \frac{d p_{1}}{d t} &=c_{1} p_{1}\left(1-p_{1}\right)-m_{1} p_{1} \\ \frac{d p_{2}}{d t} &=c_{2} p_{2}\left(1-p_{1}-p_{2}\right)-m_{2} p_{2}-c_{1} p_{1} p_{2} \end{aligned}$$ $$\begin{array}{l}{\text { where } p_{i} \text { is the fraction of patches occupied by species } i} \\ {\text { and } c_{i} \text { and } m_{i} \text { are the species-specific rates of colonization }} \\ {\text { and extinction of patches, respectively. These equations }} \\ {\text { assume that any patch has at most one species, and spe- }} \\ {\text { cies } 2 \text { patches can be taken over by species } 1, \text { but not vice }} \\ {\text { versa. }}\end{array}$$ $$\begin{array}{l}{\text { (a) Suppose that } m_{1}=m_{2}=3, c_{1}=5, \text { and } c_{2}=30 . \text { Find }} \\ {\text { all equilibria. }} \\ {\text { (b) Calculate the Jacobian matrix. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Determine the local stability properties of each equilib- }} \\ {\text { rium found in part (a) using the Jacobian from part (b). }} \\ {\text { (d) Are the species predicted to be able to coexist at a stable }} \\ {\text { equilibrium? }}\end{array}$$

Step-by-Step Solution

Answer

Stable coexistence is not possible as all equilibria indicate instability or saddle points.

Step 1: Simplify Equations with Given Parameters

Let's use the given values for \( m_1 = m_2 = 3 \), \( c_1 = 5 \), and \( c_2 = 30 \) in the model equations. The system becomes: \( \frac{d p_1}{d t} = 5p_1(1 - p_1) - 3p_1 \) and \( \frac{d p_2}{d t} = 30p_2(1 - p_1 - p_2) - 3p_2 - 5p_1 p_2 \). Simplifying further: \( \frac{d p_1}{d t} = 2p_1(1 - p_1) \) and \( \frac{d p_2}{d t} = (27 - 5p_1)p_2 - 30p_2^2 \).

Step 2: Find Equilibria

Set \( \frac{d p_1}{d t} = 0 \) and \( \frac{d p_2}{d t} = 0 \). For \( p_1 \), we get \( 2p_1(1 - p_1) = 0 \), so \( p_1 = 0 \) or \( p_1 = 1 \). For \( p_2 \) with \( p_1 = 0 \), solve \( 30p_2(1 - p_2) - 3p_2 = 0 \), getting \( p_2 = 0 \) or \( p_2 = \frac{9}{10} \). For \( p_2 \) with \( p_1 = 1 \), solve \( -5p_2 = 0 \), getting \( p_2 = 0 \). Thus, equilibria are \((0, 0)\), \((1, 0)\), and \((0, \frac{9}{10})\).

Step 3: Calculate Jacobian Matrix

For the system \( \frac{d p_1}{d t} = 2p_1(1 - p_1) \) and \( \frac{d p_2}{d t} = (27 - 5p_1)p_2 - 30p_2^2 \), the Jacobian matrix is\[J = \begin{bmatrix}\frac{\partial}{\partial p_1} [2p_1(1 - p_1)] & \frac{\partial}{\partial p_2} [2p_1(1 - p_1)] \\frac{\partial}{\partial p_1} [(27 - 5p_1)p_2 - 30p_2^2] & \frac{\partial}{\partial p_2} [(27 - 5p_1)p_2 - 30p_2^2]\end{bmatrix}\]Evaluating partial derivatives gives: \( J = \begin{bmatrix} 2 - 4p_1 & 0 \ -5p_2 & 27 - 5p_1 - 60p_2 \end{bmatrix} \).

Step 4: Analyze Stability of Equilibria

For each equilibrium:1. At \((0, 0)\), the Jacobian simplifies to \( J(0,0) = \begin{bmatrix} 2 & 0 \ 0 & 27 \end{bmatrix} \), with eigenvalues \(2\) and \(27\) (both positive), suggesting instability.2. At \((1, 0)\), \( J(1,0) = \begin{bmatrix} -2 & 0 \ 0 & 22 \end{bmatrix} \), with eigenvalues \(-2\) and \(22\) (one positive, one negative), suggesting this is a saddle point.3. At \((0, \frac{9}{10})\), \( J(0, \frac{9}{10}) = \begin{bmatrix} 2 & 0 \ -\frac{9}{2} & -27 \end{bmatrix} \), with eigenvalues \(2\) and \(-27\) (one positive, one negative), suggesting this is also a saddle point.

Step 5: Conclude on Species Coexistence

From the equilibrium and stability analysis, no stable equilibria support coexistence, suggesting the species cannot coexist at a stable equilibrium.

Key Concepts

Differential Equations

Differential equations are fundamental tools in modeling dynamic systems, displaying how certain quantities change over time. In the context of competition-colonization models, they help us quantify the change in occupation of patches by different species in a metapopulation. The equations given in the problem describe how the fraction of patches, denoted as \( p_1 \) and \( p_2 \), occupied by either species evolves.

For species 1, the change in its occupied patch fraction over time is represented by the equation \( \frac{d p_1}{d t} = c_1 p_1(1 - p_1) - m_1 p_1 \). This expression considers the colonization rate \( c_1 \) and the extinction rate \( m_1 \) of species 1. The term \( c_1 p_1(1 - p_1) \) shows growth potential, assuming more patches can be colonized until all patches are filled (hence \( 1 - p_1 \)).

• Colonization contributes positively to the fraction occupied as it increases the occupied patches.
• Extinction reduces it as some occupied patches become unoccupied over time.

Species 2’s equation \( \frac{d p_2}{d t} = c_2 p_2(1 - p_1 - p_2) - m_2 p_2 - c_1 p_1 p_2 \) is more complex due to the added factor \( c_1 p_1 p_2 \), which represents competitive displacement by species 1, making it different from species 1's equation. Here, the dynamics consider not only its colonization and extinction but also the impact of being displaced by species 1.

Species Coexistence

Species coexistence refers to the ability of two or more species to occupy the same area simultaneously without outcompeting each other to extinction. In our model, the focus is on checking whether the two species can maintain stable patches in a shared space.

To understand coexistence, we set the rates of change \( \frac{d p_1}{d t} \) and \( \frac{d p_2}{d t} \) to zero, looking for steady states where no net change occurs over time. These states are called equilibria.

• An equilibrium like \((0, 0)\) means neither species occupies any patches.
• \((1, 0)\) shows species 1 fully occupying patches, with species 2 absent.
• \((0, \frac{9}{10})\) indicates species 2 mostly occupies patches with species 1 absent.

The goal of analyzing these equilibria is to determine under what conditions both species can persist. However, the analysis shows that all equilibria are unstable or saddle points, indicating that coexistence through stable equilibrium isn't achieved in the given conditions.

Equilibrium Stability

Equilibrium stability refers to the nature of equilibria found when the rates of changes are zero. It examines whether small disturbances will return to the original equilibrium or cause a shift to a different state. Stability can be determined using techniques like perturbation analysis and often involves checking the sign of eigenvalues derived from a matrix called the Jacobian.

In the competition-colonization model, once we identify potential equilibria like \((0, 0)\), \((1, 0)\), and \((0, \frac{9}{10})\), we analyze stability by calculating the Jacobian matrix and determining its eigenvalues at these points.

If an eigenvalue is positive, then the equilibrium is unstable; small changes can cause the population to move away from the equilibrium. If all eigenvalues are negative, the equilibrium is stable, meaning it returns to the point after disturbances. However, finding an equilibrium to be a saddle point (mixed signs in eigenvalues) implies that stability is conditional, often leading to instability under general conditions. This analysis in the problem shows that no stable coexisting equilibrium exists for the given species, highlighting the instability of the arrangements.

Jacobian Matrix

The Jacobian matrix is a mathematical tool that helps analyze the stability of equilibria in systems described by differential equations. In this context, it contains partial derivatives of the system equations with respect to each variable, representing how sensitive the system is to changes in populations \( p_1 \) and \( p_2 \).

The Jacobian of our system is:
\[J = \begin{bmatrix}\frac{\partial}{\partial p_1} \left[ 2p_1(1 - p_1) \right] & \frac{\partial}{\partial p_2} \left[ 2p_1(1 - p_1) \right] \\frac{\partial}{\partial p_1} \left[ (27 - 5p_1)p_2 - 30p_2^2 \right] & \frac{\partial}{\partial p_2} \left[ (27 - 5p_1)p_2 - 30p_2^2 \right]\end{bmatrix}\]
This matrix is used to evaluate stability by examining eigenvalues. Negative eigenvalues suggest the system returns to an equilibrium point after a disturbance, indicating stability. Conversely, positive eigenvalues mean any disturbance might lead to an increase or decrease, moving the system away from equilibrium, showing instability.

• A key strategy is classifying the nature of equilibria based on the Jacobian.
• It provides insight into long-term behaviors and potential coexistence or dominance scenarios in the metapopulation.

For the given problem, the calculations revealed conditions under which the species could not stably coexist, with the Jacobian matrix's eigenvalues confirming the instability at potential equilibria points.