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Dynamics of a Ratio-Dynamics of a Ratio-Dependent Predator-Prey Dependent Predator-Prey Model with Nonconstant Model with Nonconstant
Harvesting PoliciesHarvesting PoliciesCatherine Lewis and Benjamin Leard
August 1st, 2007
Predator-Prey ModelsPredator-Prey Models• 1925 & 1926: Lotka and Volterra independently
propose a pair of differential equations that model the relationship between a single predator and a single prey in a given environment:
x rx axy
y bxy cy
Variable and Parameter definitionsx – prey species populationy – predator species populationr – Intrinsic rate of prey population Increasea – Predation coefficientb – Reproduction rate per 1 prey eatenc – Predator mortality rate
Ratio-Dependent Predator-Prey Ratio-Dependent Predator-Prey ModelModel
(1 )axy
x x xy x
bxyy dy
y x
Parameter/Variable Definitionsx – prey populationy – predator populationa – capture rate of preyd – natural death rate of predatorb – predator conversion rate
Prey growth term Predation term
Predator death term Predator growth term
First GoalFirst Goal
( )H x hx
( )hx
H xc x
Analyze the model with two non-constant harvesting functions in the prey equation.
1.
2.
Third GoalThird GoalFind bifurcations, periodic orbits, and
connecting orbits.
Example of Hopf Bifurcation
Logistic Equation Bifurcation Diagram
Model One: Constant Effort Model One: Constant Effort HarvestingHarvesting
(1 )axy
x x x hxy x
bxy y d
y x
• The prey is harvested at a rate defined by a linear function.•Two equilibria exist in the first quadrant under certain parameter values.• One of the points representscoexistence of the species.•Maximum Harvesting Effort = 1
Model Two: Limit HarvestingModel Two: Limit Harvesting
(1 )axy hx
x x xy x c x
bxy y d
y x
• The prey is harvested at a rate defined by a rational function.• The model has three equilibria that exist in the first quadrant under certain conditions.• Again, one of the points represents coexistence of the species.
ConclusionsConclusions
• Coexistence is possible under both harvesting policies.
• Multiple bifurcations and connecting orbits exist at the coexistence equilibria.
• Calculated Maximum Sustainable Yield for model one.
Future ResearchFuture Research
• Study ratio-dependent models with other harvesting policies, such as seasonal harvesting.
• Investigate the dynamics of harvesting on the predator species or both species.
• Study the model with a harvesting agent who wishes to maximize its profit.