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6.3 Ecological Models (continued)

last time: predator-prey

\[ x' = x(a - py) \]\[ y' = y(-b + qx) \]

prey

predator (uses prey as food)

today: competition system

two species going after the same food source but not each other (squirrels and chipmunks)

\[ \frac{dx}{dt} = a_1x - b_1x^2 - c_1xy = x(a_1 - b_1x - c_1y) \]\[ \frac{dy}{dt} = a_2y - b_2y^2 - c_2xy = y(a_2 - b_2y - c_2x) \]

\( a_i, b_i, c_i > 0 \)

if \( y \) is not present,

\[ x' = a_1x - b_1x^2 = x(a_1 - b_1x) \]

logistic growth

grow until carrying capacity \( x = \frac{a_1}{b_1} \)

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both \( x, y \) grow logistically

the presence of the other reduces the rate and reduces carrying capacity

A graph of population vs time showing two curves approaching a horizontal dashed line at a1/b1.

example

\[ x' = x(1 - x - y) \]\[ y' = y(\frac{3}{4} - y - \frac{1}{2}x) \]

cp: \( (0,0), (1,0), (0, \frac{3}{4}), (\frac{1}{2}, \frac{1}{2}) \)

both die
y dies
x dies
coexistence

Given \( x(0), y(0) \), what happens if \( t \to \infty \)

\[ J(x,y) = \begin{bmatrix} 1 - 2x - y & -x \\ -\frac{1}{2}y & \frac{3}{4} - 2y - \frac{1}{2}x \end{bmatrix} \]
\[ J(0,0) = \begin{bmatrix} 1 & 0 \\ 0 & \frac{3}{4} \end{bmatrix} \quad \lambda = 1, \frac{3}{4} \quad \text{source, unstable} \]
\[ \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
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Stability Analysis of Critical Points

For the critical point at \( (1,0) \):

\[ J(1,0) = \begin{bmatrix} -1 & -1 \\ 0 & -\frac{1}{4} \end{bmatrix} \quad \lambda = -1, -\frac{1}{4} \quad \text{Saddle, unstable} \]\[ \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 4 \\ -3 \end{bmatrix} \]

For the critical point at \( (0, \frac{3}{4}) \):

\[ J(0, \frac{3}{4}) = \begin{bmatrix} \frac{1}{4} & 0 \\ -\frac{9}{16} & -\frac{3}{4} \end{bmatrix} \quad \lambda = \frac{1}{4}, -\frac{3}{4} \quad \text{Saddle, unstable} \]\[ \vec{v} = \begin{bmatrix} 8 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]

For the critical point at \( (\frac{1}{2}, \frac{1}{2}) \):

\[ J(\frac{1}{2}, \frac{1}{2}) = \begin{bmatrix} -\frac{1}{2} & -\frac{1}{2} \\ -\frac{3}{8} & -\frac{1}{2} \end{bmatrix} \quad \lambda \approx -0.146, -0.854 \quad \text{Sink, asymp. stable} \]\[ \vec{v} = \begin{bmatrix} \sqrt{2} \\ -1 \end{bmatrix}, \begin{bmatrix} \sqrt{2} \\ 1 \end{bmatrix} \]

Phase Portrait and Coexistence

The phase portrait illustrates the trajectories of the system. Any initial condition with \( x(0) \neq 0 \) and \( y(0) \neq 0 \) will lead to coexistence, representing "weak" competition.

Phase portrait in the first quadrant with axes x and y. Critical points at (0,0) source, (1,0) saddle, (0, 3/4) saddle, and (1/2, 1/2) sink. Red trajectories converge to the sink.
Figure 1: Phase portrait showing critical points and trajectories.
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Example: System Analysis

\[ x' = x(1 - x - y) \]

\[ y' = y(\frac{1}{2} - \frac{1}{4}y - \frac{3}{4}x) \]

Critical Points (cp): \( (0,0), (1,0), (0,2), (\frac{1}{2}, \frac{1}{2}) \)

Jacobian Matrix

\[ J(x,y) = \begin{bmatrix} 1 - 2x - y & -x \\ -\frac{3}{4}y & \frac{1}{2} - \frac{1}{2}y - \frac{3}{4}x \end{bmatrix} \]

At \( (0,0) \):

\[ J(0,0) = \begin{bmatrix} 1 & 0 \\ 0 & \frac{1}{2} \end{bmatrix} \quad \lambda = 1, \frac{1}{2} \quad \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \text{Source} \]

At \( (1,0) \):

\[ J(1,0) = \begin{bmatrix} -1 & -1 \\ 0 & -\frac{1}{4} \end{bmatrix} \quad \lambda = -1, -\frac{1}{4} \quad \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 4 \\ -3 \end{bmatrix} \quad \text{Sink} \]

At \( (0,2) \):

\[ J(0,2) = \begin{bmatrix} -1 & 0 \\ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix} \quad \lambda = -1, -\frac{1}{2} \quad \vec{v} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \text{Sink} \]

At \( (\frac{1}{2}, \frac{1}{2}) \):

\[ J(\frac{1}{2}, \frac{1}{2}) = \begin{bmatrix} -\frac{1}{2} & -\frac{1}{2} \\ -\frac{3}{8} & -\frac{1}{8} \end{bmatrix} \quad \lambda \approx 0.16, -0.78 \]\[ \vec{v} = \begin{bmatrix} 1 \\ -1.32 \end{bmatrix}, \begin{bmatrix} 1 \\ 0.57 \end{bmatrix} \quad \text{Saddle} \]
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Phase Portrait of "Strong" Competition

In a "strong" competition model, one species eventually dies out, making coexistence unlikely. The outcome depends heavily on the initial conditions, separated by a boundary known as the separatrix.

Dividing lines - "separatrix": divides outcome based on initial conditions.

Phase portrait in the first quadrant showing trajectories moving toward sinks at (0,2) and (1,0) from a source at (0,0) and a saddle point at (1/2, 1/2).

Key Equilibrium Points:

  • \[ (0,0) \text{ : Source} \]
  • \[ (0,2) \text{ : Sink} \]
  • \[ (1,0) \text{ : Sink} \]
  • \[ (\frac{1}{2}, \frac{1}{2}) \text{ : Saddle} \]

"Strong" competition — one dies eventually. Coexistence is unlikely.

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Competition and Cooperation Systems

\[ x' = a_1 x - b_1 x^2 - c_1 xy \]\[ y' = a_2 y - b_2 y^2 - c_2 xy \]\[ a_i, b_i, c_i > 0 \]

Coexistence is likely if \[ b_1 b_2 > c_1 c_2 \]

↑
intrinsic limits↑
interaction

Cooperation System

Same as competition except \[ c_i < 0 \] (e.g., ants and aphids).

Cooperation system: coexistence very likely, but may lead to "doomsday" scenario — both populations \( \to \infty \).