Logistic Population Growth 

Topics The equation for logistic growth in discrete time 

Topics The equation for logistic growth in discrete time The definition of density-dependent growth 

Topics The equation for logistic growth in discrete time The definition of density-dependent growth Basic properties of the model 

Topics The equation for logistic growth in discrete time The definition of density-dependent growth Basic properties of the model Strange behavior of the (discrete time) model, such as damped oscillations and chaos 

From Geometric to Logistic Growth Geometric growth Nt+1 = Nt + Nt r Background Logistic growth Summary 3 / 12 

From Geometric to Logistic Growth Geometric growth Nt+1 = Nt + Nt r Logistic growth Nt+1 = Nt + Nt rmax 1 − Nt K where • rmax is the growth rate when Nt is close to 0. • K is the carrying capacity Background Logistic growth Summary 3 / 12 

Density-dependent growth Logistic growth is an example of density-dependent growth Background Logistic growth Summary 4 / 12 

Density-dependent growth Logistic growth is an example of density-dependent growth Definition: Population growth rate is affected by population size (N). Background Logistic growth Summary 4 / 12 

Density-dependent growth Logistic growth is an example of density-dependent growth Definition: Population growth rate is affected by population size (N). Implications: Resources are limited and there is a carrying capacity. Background Logistic growth Summary 4 / 12 

Graphical depiction 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) Geometric growth Background Logistic growth Summary 5 / 12 

Graphical depiction 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) Geometric growth Logistic growth Background Logistic growth Summary 5 / 12 

Growth rate (λt = Nt+1 /Nt) 0 20 40 60 80 100 0 10 20 30 40 50 60 Time (t) Population size (N) Inflection point (K/2) 0 20 40 60 80 100 1.00 1.02 1.04 1.06 1.08 Time (t) Growth rate: (λ=Nt+1 Nt ) Background Logistic growth Summary 6 / 12 

Growth (∆t = Nt+1 − Nt) 0 20 40 60 80 100 0 10 20 30 40 50 60 Time (t) Population size (N) Inflection point (K/2) 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (t) Growth: (∆N =Nt+1 − Nt ) Background Logistic growth Summary 7 / 12 

Growth (∆t = Nt+1 − Nt) as a function of N 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Population size (N) Growth: (∆N =Nt+1 − Nt ) Inflection point (K/2) Background Logistic growth Summary 8 / 12 

What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) rmax=0.1 Background Logistic growth Summary 9 / 12 

What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) rmax=0.1 rmax=0.5 Background Logistic growth Summary 9 / 12 

What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Damped oscillation Time (t) Population size (N) rmax=0.1 rmax=0.5 rmax=2.0 Background Logistic growth Summary 9 / 12 

What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Chaos Time (t) Population size (N) rmax=0.1 rmax=0.5 rmax=2.0 rmax=3.0 Background Logistic growth Summary 9 / 12 

Definitions Overcompensation Density-dependent response in which populations over- or under-shoot carrying capacity rather than approach it gradually Background Logistic growth Summary 10 / 12 

Definitions Overcompensation Density-dependent response in which populations over- or under-shoot carrying capacity rather than approach it gradually Chaos Highly variable dynamics that are extremely sensitive to small changes in parameters Background Logistic growth Summary 10 / 12 

Assumptions of basic model • K and rmax are constant • No sex or age effects or other sources of individual heterogeneity • No time lags • No stochasticity Background Logistic growth Summary 11 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth Background Logistic growth Summary 12 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K Background Logistic growth Summary 12 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) Background Logistic growth Summary 12 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. Background Logistic growth Summary 12 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. • But it does allow for complex dynamics that resemble patterns seen in nature. Background Logistic growth Summary 12 / 12 

Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. • But it does allow for complex dynamics that resemble patterns seen in nature. Assignment Read pages 32–36 in Conroy and Carroll Background Logistic growth Summary 12 / 12