participation:

\\frac{dx}{dt} = -6x + 2xy\
\\frac{dy}{dt} = 10y - 5xy\

a) identify prey-pred species
b) classify cp
c) determine direction of travel & trajectory curve
d) define possible long-term behavior

Question

participation:

\\frac{dx}{dt} = -6x + 2xy\
\\frac{dy}{dt} = 10y - 5xy\

a) identify prey-pred species
b) classify cp
c) determine direction of travel & trajectory curve
d) define possible long-term behavior

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Lotka-Volterra Equations", "Jacobian Stability Analysis", "Phase Portrait Trajectories" ], "current_concepts": [ "Lotka-Volterra Equations", "Jacobian Stability Analysis", "Phase Portrait Trajectories" ] } ## Identify prey and predator species $$ \begin{aligned} &\frac{dx}{dt} = -6x + 2xy = x(-6 + 2y) \ &\frac{dy}{dt} = 10y - 5xy = y(10 - 5x) \ &\text{Since } \frac{dx}{dt} \text{ has a negative growth rate } (-6x) \text{ and benefits from interaction } (+2xy), \text{ } x \text{ is the predator.} \ &\text{Since } \frac{dy}{dt} \text{ has a positive growth rate } (+10y) \text{ and suffers from interaction } (-5xy), \text{ } y \text{ is the prey.} \end{aligned} $$ ## Find and classify critical points $$ \begin{aligned} &x(-6 + 2y) = 0 \implies x = 0 \text{ or } y = 3 \ &y(10 - 5x) = 0 \implies y = 0 \text{ or } x = 2 \ &\text{Critical points: } (0,0) \text{ and } (2,3) \ &J(x,y) = \begin{bmatrix} -6+2y & 2x \ -5y & 10-5x \end{bmatrix} \ &J(0,0) = \begin{bmatrix} -6 & 0 \ 0 & 10 \end{bmatrix} \implies \lambda = -6, 10 \implies \text{Saddle point} \ &J(2,3) = \begin{bmatrix} 0 & 4 \ -15 & 0 \end{bmatrix} \implies \lambda^2 + 60 = 0 \implies \lambda = \pm i\sqrt{60} \implies \text{Center} \end{aligned} $$ ## Determine trajectory direction and long-term behavior $$ \begin{aligned} &\text{At } (2, 4) \text{ (above the center): } \frac{dx}{dt} = 2(-6+8) = 4 > 0 \implies \text{moves right (clockwise)} \ &\text{Long-term behavior: Trajectories form closed periodic orbits around the center } (2,3). \end{aligned} $$ a) **Prey-predator identification:** - Prey: $y$ (has positive natural growth rate $10y$) - Predator: $x$ (has natural decay rate $-6x$) b) **Critical points classification:** - $(0,0)$: Saddle point (eigenvalues $\lambda = -6, 10$) - $(2,3)$: Center (eigenvalues $\lambda = \pm i\sqrt{60}$) c) **Trajectory direction:** - Clockwise around the center $(2,3)$ d) **Long-term behavior:** - The populations will oscillate periodically in closed orbits around the stable center $(2,3)$. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Lotka-Volterra Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify prey and predator species

Find and classify critical points

Determine trajectory direction and long-term behavior

Answer: