Cannibalism: When Parents attack (Their Kids)

Cannibalism: When Parents attack (Their Kids) By the fine young cannibals starring: Jonna Anderson, Sam Potter, and Tracy Robson Directed by: Dr. Ledder and Shannon Jessie

Biological Setting • Small single-species pond or lake. • Ecosystem consists of one fish species (for example, largemouth bass) and macroinvertebrates. • Fish diet is size-dependent. • Macroinvertebrates’ food source is considered constant.

Equation Process Macroinvertebrate (prey) population = controlled growth – predation (by juvenile population) Juvenile population = eggs + predation – recruitment – death – cannibalization Adult population = cannibalism + recruitment – death – eggs

FunctionalResponse • First model-Holling Type I • Second model-Holling Type II • Holling Type III

Equation Process Part II

parameters Ratio of rate of biomass loss of juveniles to prey growth rate (0.1) Ratio of rate of biomass loss of adults to biomass loss by juveniles (0.8) Gain from macroinvertebrate predation relative to juvenile biomass loss (4) Net efficiency of egg-laying and recruitment biomass transfer cycle (0.1) Ability of the environment to support macroinvertebrates Efficiency of egg-laying and cannibalism biomass transfer cycle in adults (0.05) Semi-saturation constant relative to carrying capacity

Analysis • Methods Used: • Known points: • Find Equilibrium Points • Determine Stability Using Jacobian and Routh-Herwitz Conditions • Unknown points: • Nullcline Analysis • Determine Necessary Conditions

Equilibrium Points (0, 0, 0)---Extinction Point (1, 0, 0)---Prey Only Lives (0, J, A)---Prey Dies, All Others Live (P, J, A)---All Three Live

General Jacobian

Routh-Hurwitz Conditions From the Jacobian matrix: Asymptotically stable if:

Stability of E point • E point: (0,0,0) • Trace: • Determinant:

Stability of P • P Point: (1,0,0) • Trace: • Determinant: • P Point: (1,0,0) • Trace: • Determinant: • Thus, condition:

Nullcline Analysis • J-A is an autonomous subsystem, where p satisfies the algebraic relation: • J-nullcline • A-nullcline • An intersection permits an equilibrium point J nullcline-solid red A nullcline-dashed blue

Nullcline analysis Derivative of J nullcline < Derivative of A nullcline at (0,0). where and and

Nullcline analysis With A and J going to 0, P goes to 1. Substituting into our inequality: Derivative of J nullcline < Derivative of A nullcline at (0,0). Which reduces to:

Graphs PJA equilibrium – stable (1, 0, 0) – stable

Graphs Limit Cycle – Stable cycle, unstable points

Stability • Old Model: Existence and stability guaranteed by: • New Model: Taking lim in the trace of the Jacobian, stability can be guaranteed with . • Increasing increases stability.

What’s next? • Prove that P, J, and A are bounded • Prove global stability of _____ • Prove that is not only sufficient, but necessary, • for a PJA equilibrium. • Find conditions under which stability is guaranteed.

Thank you to Dr. Ledder and Shannon Jessie! Questions? ¿Preguntas?

Cannibalism: When Parents attack (Their Kids)