Last time d N = [ b (1 N/k ) d ] N N = 0 or N = k (1 d/b ) = - PowerPoint PPT Presentation

Last time d N ¯ ¯ = [ b (1 − N/k ) − d ] N N = 0 or N = k (1 − d/b ) = k (1 − 1 /R 0 ) d t d N ¯ ¯ = [ b − d (1 + N/k )] N N = 0 or N = k ( b/d − 1) = k ( R 0 − 1) d t d N ¯ ¯ = rN (1 − N/K ) N = 0 or N = K d t Figures taken from NRC Handelsblad 11 Dec 2010 (left) and Wikipedia (right).

Chapter 3 Lotka Volterra model Theoretical Biology 2016

Suppose measurements for the prey p.c. mouse death rate p.c. mouse birth rate (a) (b) (c) mice eaten per owl y=b(1-R/k) y=aR b d mouse density R mouse density R mouse density R d R d t = [ bf ( R ) − d − aN ] R where f ( R ) = 1 − R/k with maximum birth rate b , death rate d , and a predation term aRN .

Suppose measurements for the prey p.c. mouse death rate p.c. mouse birth rate cells killed per killer cell tumor cell division rate (a) (b) (c) mice eaten per owl tumor cell death rate y=b(1-R/k) y=aR b d tumor cells tumor cells tumor cells mouse density R mouse density R mouse density R d R d t = [ bf ( R ) − d − aN ] R where f ( R ) = 1 − R/k with maximum birth rate b , death rate d , and a predation term aRN .

Measurements for the predator (d) (e) p.c. owl birth rate owl death rate y=caR δ aR mice eaten per owl owl density N d N d t = [ caR − δ ] N where 1/ δ is the expected owl life span

Measurements for the predator (d) (e) p.c. owl birth rate killer cell division rate killer cell death rate owl death rate y=caR δ aR tumor cells seen per killer cells mice eaten per owl owl density N killer cell d N d t = [ caR − δ ] N where 1/ δ is the expected killer cell life span

d R d t = [ bf ( R ) − d − aN ] R where f ( R ) = 1 − R/k with maximum birth rate b , death rate d , and a predation term aRN . In the abscence of predators the carrying capacity is: ¯ R = k (1 − d/b ) = k (1 − 1 /R 0 ) = K Number of predators: d N d t = [ caR − δ ] N where 1 / δ is the expected life-span. 32

Steady states Setting d R/ d t = d N/ d t = 0 yields N = 1 R = 0 and a [ b (1 − R/k ) − d ] R = δ N = 0 and ca Trivial: ( ¯ R, ¯ N ) = (0 , 0) and ( ¯ R, ¯ N ) = ( K, 0). Non-trivial: N = 1 δ ⇤ � ⇥ ⌅ ¯ b 1 − − d a cak

Nullclines in phase space (a) (b) b − d a N K K δ δ R R ca ca d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N

LV-model typically written with logistic growth (a) (b) r/a b − d a N K K δ δ R R ca ca d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N

What happens if we feed the prey? (a) (b) r/a b − d a N K K δ δ R R ca ca Increasing K in (b): 1. increases the prey density d R d N 2. increases the predator density d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N 3. increases the prey density until predators can survive

What happens if we feed the prey? (a) (b) r/a 3. increases the b − d a prey density N until predators can survive K K K K δ δ K R R ca ca d R d N d t = [ r (1 − R/K ) − aN ] R and d t = [ caR − δ ] N d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N By feeding the prey we get predators

What happens if we feed the prey? (a) (b) r/a b − d a N K K δ δ R R ca ca Increasing K in (a): 1. increases the prey density and keeps predators the same d R d N 2. increases the predator density and keeps prey the same d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N 3. increases both populations

What happens if we feed the prey? (a) (b) 2. increases the r/a predator b − d a density and N keeps prey the same K K K K δ δ R R ca ca d R d N d t = [ r (1 − R/K ) − aN ] R and d t = [ caR − δ ] N d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N By feeding the prey we get more predators

Example: bacterial food chain ← Predator Colpidium striatium ← Prey with predator Serratia marcescens ← Prey alone Serratia marcescens (b): The effect of nutrients on the density of prey (a): The same for prey (a: open circles) and a predator (a: closed circles). From: Kaunzinger et al. Nature 1998.

What happens if we change a ? (a) (b) Prey: tumor cells r/a b − d Predator: killer cells a N a : drug changing the killing rate K K δ δ R R ca ca d R d N d t = [ r (1 − R/K ) − aN ] R and d t = [ caR − δ ] N d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N

Changing the predation rate a changes both r 1 1 N N a N N r 0.5 0.5 a 0 0 R K K R = d R = d 0 0.5 1 0 0.5 1 ca ca R R d R d N d t = rR (1 − R/K ) − aRN , d t = caRN − dN .

Decreasing the predation rate increases the predator 1 1 N N N N r 0.5 0.5 a 0 0 R a R = d 0 0.5 1 0 1.5 3 ca R b d R d N d t = rR (1 − R/K ) − aRN , d t = caRN − dN . Predators with larger a have higher fitness R 0

Fitness R 0 For the prey R 0 = b/d For the predator R 0 = caR which is not a constant. δ Take the best possible circumstances, i.e., R = K and let R 0 = caK δ . ca or at R = K The prey equilibrium is at R = δ R 0 This implies that a predator with an R 0 = 2 is expected to halve its prey population.

Lotka Volterra model is very general d R d N d t = [ r (1 − R/K ) − aN ] R and d t = [ caR − δ ] N Predator-prey & host-parasite models Seals in the Waddensea infected by virus Hepatocytes infected by hepatitis Cancer cells removed by killer cells Economics: interactions industries

Lotka Volterra model is very general Many models use the Lotka Volterra equations

Increasing the killing rate decreases the killers 1 1 N N N N r 0.5 0.5 a 0 0 R a R = d 0 0.5 1 0 1.5 3 ca R b d R d N d t = rR (1 − R/K ) − aRN , d t = caRN − dN . Tumor Killer cells

For example the SARS epidemic with R 0 = 3, and β = 1 . 5 and δ = 0 . 5 per week − 10 10 9 10 d t 8 10 7 Number of people 10 6 10 d I d t = [ β − δ ] I 5 10 4 10 3 10 Susceptibles 2 Infected (Lotka Volterra) 10 Infected (I’=( β−δ )I 1 10 0 10 0 4 8 12 16 20 24 28 32 36 40 44 48 52 Time in weeks d S d I d t = rS (1 − S/K ) − β SI and d t = β SI − δ I

Alternative: prey maintained by source N d R d t = s − dN − aRN predator remains: d N d t = [ caR − δ ] N R σ K = s δ d ca

Several Lotka Volterra like models Lotka-Volterra model (with birth and death rates): d R d N d t = [ b (1 − R/k ) − d − aN ] R and d t = [ caR − δ ] N Lotka-Volterra model (with logistic growth): d R d N d t = [ r (1 − R/K ) − aN ] R and d t = [ caR − δ ] N Resource maintained by a source: d R d N d t = s − dR − aNR and d t = [ caR − δ ] N Lotka-Volterra competition equations: d N 1 d N 2 d t = r 1 N 1 (1 − N 1 /K 1 − N 2 /c 1 ) and d t = r 2 N 2 (1 − N 2 /K 2 − N 1 /c 2 )

History from Wikipedia The Lotka–Volterra predator–prey model was proposed by Alfred J. Lotka “in the theory of autocatalytic chemical reactions” in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model to "organic systems" using a plant species and a herbivorous animal species as an example, and in 1925 he utilised it to analyse predator- prey interactions in his book on biomathematics arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic independently investigated the equations in 1926.