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Dynamics of a Ratio-Dependent Predator-Prey Model with Nonconstant Harvesting Policies. Catherine Lewis and Benjamin Leard. August 1 st , 2007. Predator-Prey Models.

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dynamics of a ratio dependent predator prey model with nonconstant harvesting policies

Dynamics of a Ratio-Dependent Predator-Prey Model with Nonconstant Harvesting Policies

Catherine Lewis and Benjamin Leard

August 1st, 2007

predator prey models
Predator-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:

Variable and Parameter definitions

x – prey species population

y – predator species population

r – Intrinsic rate of prey population Increase

a – Predation coefficient

b – Reproduction rate per 1 prey eaten

c – Predator mortality rate

ratio dependent predator prey model
Ratio-Dependent Predator-Prey Model

Prey growth term

Predation term

Parameter/Variable Definitions

x – prey population

y – predator population

a – capture rate of prey

d – natural death rate of predator

b – predator conversion rate

Predator death term

Predator growth term

previous research
Previous Research

Harvestingon the Prey Species

first goal
First Goal

Analyze the model with two non-constant harvesting functions in the prey equation.

1.

2.

slide6

Second Goal

Find equilibrium points and determine local stability.

third goal
Third Goal

Find bifurcations, periodic orbits, and connecting orbits.

Logistic Equation Bifurcation Diagram

Example of Hopf Bifurcation

model one constant effort harvesting
Model One: Constant Effort Harvesting
  • 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 represents
  • coexistence of the species.
  • Maximum Harvesting Effort = 1
bifurcations in model one
Bifurcations in Model One

Hopf Bifurcation

Transcritical Bifurcation

model two limit harvesting
Model Two: Limit Harvesting
  • 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.
bifurcations in model two
Bifurcations in Model Two

Bifurcation Diagram

conclusions
Conclusions
  • 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 research
Future 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.