Models of interspecific interactions Predator-prey dynamics and competition 

Introduction • Lotka and Volterra developed models for both predator-prey dynamics and competitive interactions Introduction Predator-Prey Competition 2 / 14 

Introduction • Lotka and Volterra developed models for both predator-prey dynamics and competitive interactions • As usual, these models were developed as continuous-time models Introduction Predator-Prey Competition 2 / 14 

Introduction • Lotka and Volterra developed models for both predator-prey dynamics and competitive interactions • As usual, these models were developed as continuous-time models • We will focus on discrete-time versions (t = 1, 2, . . .) Introduction Predator-Prey Competition 2 / 14 

Introduction • Lotka and Volterra developed models for both predator-prey dynamics and competitive interactions • As usual, these models were developed as continuous-time models • We will focus on discrete-time versions (t = 1, 2, . . .) • We will ignore potential extensions with stochasticity, age structure, spatial structure, etc. . . Introduction Predator-Prey Competition 2 / 14 

Question How should predator-prey dynamics operate? Introduction Predator-Prey Competition 3 / 14 

Question How should predator-prey dynamics operate? Introduction Predator-Prey Competition 3 / 14 

Lynx-hare cycles Introduction Predator-Prey Competition 4 / 14 

Lotka-Volterra predator-prey model Model for prey Nprey t+1 = Nprey t + Nprey t (rprey − dpreyNpred t ) Introduction Predator-Prey Competition 5 / 14 

Lotka-Volterra predator-prey model Model for prey Nprey t+1 = Nprey t + Nprey t (rprey − dpreyNpred t ) Model for predator Npred t+1 = Npred t + Npred t (bpredNprey t − dpred) Introduction Predator-Prey Competition 5 / 14 

Lotka-Volterra predator-prey model Model for prey Nprey t+1 = Nprey t + Nprey t (rprey − dpreyNpred t ) Model for predator Npred t+1 = Npred t + Npred t (bpredNprey t − dpred) • Model is based on geometric growth • rprey is the growth rate of the prey in the absence of predators • dprey is the predation rate • bpred is the birth rate of the predators • dpred is the mortality rate of the predator Introduction Predator-Prey Competition 5 / 14 

Equilibrium Equilibrium for prey occurs when. . . Npred = rprey dprey Introduction Predator-Prey Competition 6 / 14 

Equilibrium Equilibrium for prey occurs when. . . Npred = rprey dprey Equilibrium for predators occurs when. . . Nprey = dpred bpred Introduction Predator-Prey Competition 6 / 14 

Equilibrium Equilibrium for prey occurs when. . . Npred = rprey dprey Equilibrium for predators occurs when. . . Nprey = dpred bpred However, it is rare that both equilibrium conditions will be met at the same time, and so the populations will cycle. Introduction Predator-Prey Competition 6 / 14 

Model predicts population cycles qqqqqqqqqqqqqqqqqqqq q q q q qqqqqq q q q q q q q qq q q q q q q q qqq q q q q q q q q q q q qq q q q q qqqqqqqqqq 0 20 40 60 0 50 100 150 200 250 Time Population size qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qqq q q q q qqqqqqqqqqq q q q q q q q q q q qqqqqqq q q Prey Predator Introduction Predator-Prey Competition 7 / 14 

Isle Royale wolves and moose https://youtu.be/PdwnfPurXcs Introduction Predator-Prey Competition 8 / 14 

Competition David Bygott Introduction Predator-Prey Competition 9 / 14 

Lotka-Volterra competition model Model for species A NA t+1 = NA t + rANA t (KA − NA t − αBNB t )/KA Introduction Predator-Prey Competition 10 / 14 

Lotka-Volterra competition model Model for species A NA t+1 = NA t + rANA t (KA − NA t − αBNB t )/KA Model for species B NB t+1 = NB t + rBNB t (KB − NB t − αANA t )/KB Introduction Predator-Prey Competition 10 / 14 

Lotka-Volterra competition model Model for species A NA t+1 = NA t + rANA t (KA − NA t − αBNB t )/KA Model for species B NB t+1 = NB t + rBNB t (KB − NB t − αANA t )/KB • Model based on logistic growth • The α parameters are competition coefficients determining how strongly each species affects the other Introduction Predator-Prey Competition 10 / 14 

Equilibrium Equilibrium for species A NA = KA − αBKB 1 − αAαB Introduction Predator-Prey Competition 11 / 14 

Equilibrium Equilibrium for species A NA = KA − αBKB 1 − αAαB Equilibrium for species B NB = KB − αAKA 1 − αAαB Introduction Predator-Prey Competition 11 / 14 

Three possible outcomes Outcomes depend on the sign of the numerators (1) Stable coexistence (2) Competitive exclusion (3) Unstable equilibrium Introduction Predator-Prey Competition 12 / 14 

Three possible outcomes Outcomes depend on the sign of the numerators (1) Stable coexistence (2) Competitive exclusion (3) Unstable equilibrium Competitive exclusion principle: Two species with the same niche cannot coexist on the same limiting resource Introduction Predator-Prey Competition 12 / 14 

Outcomes q q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 20 40 60 0 50 100 150 200 Stable coexistence Time Population size qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q Species A Species B q q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 20 40 60 0 50 100 150 200 Competitive exclusion Time Population size qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q Species A Species B Introduction Predator-Prey Competition 13 / 14 

Summary Predator-prey model is extension of geometric growth • Predators and prey limit each other’s growth potential Introduction Predator-Prey Competition 14 / 14 

Summary Predator-prey model is extension of geometric growth • Predators and prey limit each other’s growth potential Competition model is extension of logistic growth • Competitors influence each other’s density-dependent regulation process Introduction Predator-Prey Competition 14 / 14 

Summary Predator-prey model is extension of geometric growth • Predators and prey limit each other’s growth potential Competition model is extension of logistic growth • Competitors influence each other’s density-dependent regulation process These models could be extended to include: • More species • Stochasticity • Age structure • Harvest • Spatial structure • Additional forms of density dependence Introduction Predator-Prey Competition 14 / 14