8.2 Nonlinear EPs the Deer Moose competition model

Nonlinear Systems and Equilibrium Points

• Nonlinear systems can have multiple equilibrium points, unlike linear systems which have only one equilibrium point at (0, 0).

Example: Deer and Moose Competition Model

• The model is derived from ecology and describes competition between two species: deer (D) and moose (M).

• Both species compete for limited resources but do not eat each other.

Differential Equations

• Deer Equation (D'): The change in deer population.

  • Birthrate of deer: 3 per year (per capita).

  • Negative terms due to competition:

    • Competition with moose: 1 * D * M.

    • Intraspecies competition: 1 * D^2.

  • Therefore, D' = 3D - DD - DM.

• Moose Equation (M'): The change in moose population.

  • Birthrate of moose: 2M (assuming lower birthrate than deer).

  • Negative terms due to competition:

    • Competition with deer: 0.5 * D * M.

    • Intraspecies competition: M^2.

  • Therefore, M' = 2M - 0.5DM - MM.

Equilibrium Points

• Equilibrium occurs when D' = 0 and M' = 0.

• Solving equations:

  1. (0, 0): Both populations are zero.

  2. (0, 2): No deer, two moose exist.

  3. (3, 0): Three deer exist, no moose.

  4. (2, 1): Two deer and one moose coexist.

Stability of Equilibrium Points

• Methods used to assess stability:

  • Nullclines: Sets of points where D' = 0 or M' = 0.

  • Plotting: Analyze regions around equilibrium for direction of populations.

Observations from Nullclines

• Examine the arrows’ direction in the sectors:

  • Sector behavior indicates stable or unstable points based on arrows moving toward or away from the point.

• Stability Results:

  • (0, 0): Unstable.

  • (0, 2): Unstable.

  • (3, 0): Unstable.

  • (2, 1): Stable (coexistence possible).

Conclusion

• Stable coexistence between deer and moose expected when populations reach (2, 1).

• Mathematical tools used: nullclines and extreme value test points to analyze equilibrium stability.