Species Competition Background

• Two squirrel species in Britain: red and gray
• Gray squirrels have supplanted red squirrels, pushing them north to Scotland and parts of North Wales
• Most UK squirrels now gray due to competitive displacement
• Mathematical interaction equations model population dynamics

Population Dynamics Equations

• Time-based population change equations for both species
• Gray squirrel population (G) and red squirrel population (R) variables
• Space excluded for simplification - considering island model
• Both species undergo binary competition within their populations
• Population growth proportional to existing population size
• Competition limits population (resources, partners, homes, nuts)
• Inter-species competition terms added:
  • Alpha_gr: gray-red competition effect on gray squirrels
  • Beta_gr: gray-red competition effect on red squirrels
  • Always negative effects (competition never helps)
  • Alpha and beta magnitudes determine competitive ability

Null Cline Analysis Method

• Null clines: curves where populations aren't changing (derivatives = 0)
• Visual representation more effective than pure mathematics
• Pictures worth "thousand words" for policy makers and parliament
• Four null cline conditions:
  • G = 0 (gray population zero)
  • G = 1 - Alpha*R (gray equilibrium line)
  • R = 0 (red population zero)
  • G = (1-R)/Beta (red equilibrium line)

Phase Plane Visualization

• Red population on x-axis, gray population on y-axis
• Negative slope lines (as R increases, G decreases)
• Intersection points determine equilibrium outcomes
• Four possible crossing patterns with different biological meanings

Case 1: Gray Dominance

• Gray null cline furthest out from origin
• Three equilibrium points:
  • (0,0): both extinct (unstable)
  • Red extinct, gray survives (stable attractor)
  • Gray extinct, red survives
• All trajectories pushed toward gray survival
• Gray squirrels out-competing red squirrels
• Matches current UK situation

Case 2: Red Dominance

• Red null cline furthest out from origin
• Red squirrels dominate system
• Trajectories tend toward red survival, gray extinction
• Not observed in UK reality

Case 3: Coexistence (Small Competition)

• Null clines cross creating fourth equilibrium point
• Stable coexistence possible
• Small Alpha and Beta values (minimal competition)
• Species can "live in harmony" - separate resource niches
• Trajectories converge to stable coexistence point

Case 4: Competitive Exclusion (High Competition)

• Large competition coefficients
• Fourth equilibrium point becomes unstable
• Outcome depends on initial conditions
• Either red or gray wins based on starting populations
• "Winner takes all" scenario

Management Implications

• Current UK situation: Case 1 (gray dominance)
• Potential interventions to change dynamics:
  • Increase red competitive ability
  • Reduce inter-species competition
  • Separate resource bases (e.g., "grays love sausages, reds love peanut butter")
  • Move system toward coexistence (Case 3)

Mathematical Modeling Value

• Provides rigorous framework beyond intuitive observations
• Identifies full strategy space of possible outcomes
• Shows pathways between different scenarios
• Similar to pandemic modeling - data shows current state, math shows intervention routes
• Generalizes beyond specific mechanisms (hunting ability) to competition parameters