Predator-Prey Population Cycles: Analysis and Insights
Predator-Prey Population
Cycles
Jack Sinclair & Shane Moore
Linearization of the Lemming-Stoat
Model
➔
Linearize at ten years (t=10) and the
fixed point (lemming,stoat)=(x,y)=(10
-
1
,10
-2.5
)
➔
Complex eigenvalues and is
therefore periodic
Lemming-Stoat Model
Input Average Values
Partially Derive the Model (Lemming)
Partially Derive the Model (Stoat)
Find Eigenvalues
The point (x,y)=(10
-1
,10
-2.5
) with t=10 produces the following Jacobian
Matrix
Which yield the complex eigenvalues:
{-1.5855 - 13.0678·î, -1.5855 + 13.0678·î}Sensitivity Testing
- Individually increase and decrease each parameter to see its effect
on cycle period.
- Bifurcation value (𝛕 = 0.2) for maturation delay.
Additional Predator-Prey Systems
●
Hare-lynx system
○
Similar system of differential equations
○
Only parameter for maturation delay remained significant
○
Maturation delay value (𝛕 = 1.5)
●
Moose-wolf system
○
Wolf maturation delay time of 1.8 years estimates a 38 year
population cycle period.
○
Falls in line with past estimates and observations.
Bifurcation
- Threshold values of maturation
delay which differ for each
predator-prey system.
- No population cycles in cases a,
c, and d.
- Periodic populations in case b.
Conclusion
- Oscillating population cycles corroborated through an analysis of
the linearized system.
- Maturation delay of the predator species is a key determinant for
period lengths of population cycles.
- Bifurcation values of the maturation delay parameter signal changes
in the predator-prey population relationship.
Explore the dynamics of predator-prey population cycles through mathematical models, linearization techniques, sensitivity testing, and conclusions on the role of maturation delay in shaping cycle periods and relationships between species.
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Presentation Transcript
Predator-Prey Population Cycles Jack Sinclair & Shane Moore
Linearization of the Lemming-Stoat Model Linearize at ten years (t=10) and the fixed point (lemming,stoat)=(x,y)=(10- 1,10-2.5) Complex eigenvalues and is therefore periodic
Find Eigenvalues The point (x,y)=(10-1,10-2.5) with t=10 produces the following Jacobian Matrix Which yield the complex eigenvalues: {-1.5855 - 13.0678 , -1.5855 + 13.0678 }
Sensitivity Testing - Individually increase and decrease each parameter to see its effect on cycle period. - Bifurcation value (? = 0.2) for maturation delay.
Additional Predator-Prey Systems Hare-lynx system Similar system of differential equations Only parameter for maturation delay remained significant Maturation delay value (? = 1.5) Moose-wolf system Wolf maturation delay time of 1.8 years estimates a 38 year population cycle period. Falls in line with past estimates and observations.
Bifurcation - Threshold values of maturation delay which differ for each predator-prey system. - No population cycles in cases a, c, and d. - Periodic populations in case b.
Conclusion - Oscillating population cycles corroborated through an analysis of the linearized system. - Maturation delay of the predator species is a key determinant for period lengths of population cycles. - Bifurcation values of the maturation delay parameter signal changes in the predator-prey population relationship.
