9/30 Predators and Prey

Hawk-Dove Model Overview

• Basic Concepts

  • The Hawk-Dove model is used in behavioral ecology to analyze strategies animals use when competing for resources, such as food or territory.

  • Two main strategies: Hawks and Doves.

Definitions and Explanations

• Hawks:

  • Aggressive individuals that fight for resources.

  • They can win or lose encounters with other hawks but incur costs when injured.

• Doves:

  • Non-aggressive individuals that retreat from confrontation.

  • Avoids the cost of injuries by not engaging in fights.

• Resource Value ($v$): The worth of the resource being fought over.

• Cost of Injury ($c$): The cost incurred from a fight, often exceeding the value of the resource.

Interactions and Payoffs

• When two hawks interact:

  • Payoff for each:
    $ext{Payoff} = rac{v - c}{2}$ (50% chance of winning the resource, 50% chance of losing and incurring injury).

• When a hawk meets a dove:

  • The hawk wins and receives the full value of the resource:
    $ext{Payoff for Hawk} = v \quad ( ext{Payoff for Dove} = 0)$

• When doves meet:

  • Both doves share the resource, yielding:
    $ext{Payoff for Dove} = rac{v}{2}$

Optimal Strategies

• Evolutionarily Stable Strategy (ESS):

  • A strategy is ESS if it can maintain its prevalence within a population despite the introduction of other strategies.

• A population of only hawks will not sustain doves, as doves will face zero payoff while hawks incur negative payoffs.

• Payoff Comparison:

  • Hawk vs. Hawk: Payoff diminishes under heavy costs of injury compared to resource value.

  • Important conditions are when:
    v > c for hawks to have advantageous payoffs.

Equilibrium Calculation

• Given different resource values and injury costs, we derive proportions of hawks and doves that can coexist.

• Equilibrium found by equating payoffs:

  • Hawk Payoff formula: $(-1) imes p^2 + 3 imes p(1-p)$

  • Dove Payoff formula: $0 imes p + 1.5 imes (1-p)^2$

• For example, if $v = 3$ and $c = 5$, results in:

  • Equilibrium occurs at about 60% hawks and 40% doves in a stable population.

Figures and Graphs

• Graphical Representation:

  • The crossover point of the payoff lines illustrates the ESS in the population dynamic

  • The population stabilizes around a 60/40 ratio of hawks to doves given specific resource values and fight costs.

Behavioral Application in Real-life Scenarios

• Example With Butterflies:

  • In species like the Great Eggfly butterflies, territory holders tend to fight more aggressively for their established territories compared to newcomers.

  • The experience of holding territories affects fighting persistence more than raw aggression.

Additional Considerations and Layers

• Dear Enemy Effect:

  • A phenomenon observed in territorial animals where neighbors engage in ritualized confrontations rather than full-blown fights, leading to stability in territory holding.

• Defender Strategy:

  • Addition of a third strategy defined as “Defender”, where individuals show dual behavior based on ownership—acting as hawks when defending their territory and as doves otherwise.

Evolutionary Implications

• Analyzed under different contexts, this model helps predict behaviors and strategies in various animal species.

• The distinction between hawk, dove, and defender strategies provides a framework for understanding competition, resource allocation, and mating success in different species.

Conclusion

• The Hawk-Dove model demonstrates complex dynamics of animal behaviors, illustrating how interactions can influence evolutionary outcomes in populations.

• Current research can build upon these models to provide insight into animal ecology and behavior within specific ecologies or habitats.