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6.3 Ecological Models (continued)

Last time: predator-prey

\[\begin{aligned} x' &= x(a - py) \\ y' &= y(-b + gx) \end{aligned}\]

y uses x as food

Competition system: two species go after a common food

but not each other (e.g. squirrels and chipmunks)

\[\begin{aligned} x' &= a_1 x - b_1 x^2 - c_1 xy = x(a_1 - b_1 x - c_1 y) \\ y' &= a_2 y - b_2 y^2 - c_2 xy = y(a_2 - b_2 y - c_2 x) \end{aligned}\]

\( a_i, b_i, c_i > 0 \)

In the absence of y, \( x' = x(a_1 - b_1 x) \)

logistic growth: grow until the carrying capacity then stabilizes

introduction of y slows down growth and lowers carrying capacity (same for y)

Figure: Graph of population x over time t showing logistic growth curves approaching a carrying capacity K/b.

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Example

\[\begin{aligned} x' &= x(1 - x - y) \\ y' &= y\left(\frac{3}{4} - y - \frac{1}{2}x\right) \end{aligned}\]

Critical Points (cp):

\((0, 0)\): both die
\((1, 0)\): y dies
\((0, \frac{3}{4})\): x dies
\((\frac{1}{2}, \frac{1}{2})\): coexistence

Jacobian Matrix

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

\( J(0,0) = \begin{bmatrix} 1 & 0 \\ 0 & \frac{3}{4} \end{bmatrix} \)

\( \lambda = 1, \frac{3}{4} \) | \( \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) | source, unstable

\( J(1,0) = \begin{bmatrix} -1 & -1 \\ 0 & \frac{1}{4} \end{bmatrix} \)

\( \lambda = -1, \frac{1}{4} \) | \( \vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 4 \\ -5 \end{bmatrix} \) | saddle, unstable

\( J(0, \frac{3}{4}) = \begin{bmatrix} \frac{1}{4} & 0 \\ -\frac{3}{8} & -\frac{3}{4} \end{bmatrix} \)

\( \lambda = \frac{1}{4}, -\frac{3}{4} \) | \( \vec{v} = \begin{bmatrix} 8 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) | saddle

\( J(\frac{1}{2}, \frac{1}{2}) = \begin{bmatrix} -\frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{4} & -\frac{1}{2} \end{bmatrix} \)

\( \lambda \approx -0.15, -0.85 \) | \( \vec{v} \approx \begin{bmatrix} \sqrt{2} \\ -1 \end{bmatrix}, \begin{bmatrix} \sqrt{2} \\ 1 \end{bmatrix} \) | sink, asymp. stable

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Unless either \(x(0)\), \(y(0)\) or both are 0, coexistence is the eventual outcome.

↓

"Weak" Competition

Both settle down at levels below their intrinsic carrying capacities.

Figure: Phase portrait in the first quadrant with axes x and y. Trajectories flow away from (0,0) and toward (1/2, 1/2).

Key Equilibrium Points:

\((0, 0)\): Source
\((1, 0)\): Saddle
\((0, 3/4)\): Saddle
\((1/2, 1/2)\): Sink

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Example

\[ \begin{aligned} x' &= x(1 - x - y) \\ y' &= y\left(\frac{1}{2} - \frac{1}{4}y - \frac{3}{4}x\right) \end{aligned} \]

Critical points (cp): \((0, 0), (1, 0), (0, 2), (1/2, 1/2)\)

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

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 \((1/2, 1/2)\):

\[ J(1/2, 1/2) = \begin{bmatrix} -1/2 & -1/2 \\ -3/8 & -1/8 \end{bmatrix} \quad \lambda \approx 0.16, -0.78 \quad \vec{v} \approx \begin{bmatrix} 1 \\ -1.3 \end{bmatrix}, \begin{bmatrix} 1 \\ 0.6 \end{bmatrix} \quad \text{Saddle} \]

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Competition Models and Phase Portraits

The phase portrait below illustrates a system of "strong" competition. In this scenario, the eventual outcome is the extinction of one species, depending on the initial conditions.

A "separatrix" (indicated by the red curves) divides the regions where initial conditions lead to different outcomes.

(0,0) Source: The origin acts as a source.
(0,2) Sink: A stable equilibrium point on the y-axis.
(1,0) Sink: A stable equilibrium point on the x-axis.
(½, ½) Saddle: An unstable equilibrium point where trajectories are pushed away towards one of the sinks.

Figure: Phase portrait in the first quadrant with axes x and y, showing trajectories and equilibria for a competition model.

System Equations

\[ \begin{aligned} x' &= a_1 x - b_1 x^2 - c_1 xy \\ y' &= a_2 y - b_2 y^2 - c_2 xy \end{aligned} \]\[ a_i, b_i, c_i > 0 \]

Coexistence Condition:

Coexistence is likely ("weak" competition) if:

\[ b_1 b_2 > c_1 c_2 \]

In this case, intrinsic limits are more important than interactions.

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Cooperation Systems

If \( c_i < 0 \) → cooperation system (e.g., ants and aphids).

Interaction boosts the carrying capacities.
Coexistence is most likely.
But may lead to a "doomsday" scenario → when both \( x \) and \( y \) go to \( \infty \).