d N d t = [ bf ( N ) dg ( N )] N Populations change by immigration, - - PowerPoint PPT Presentation

Chapter 3: Density dependence

(a) (b)

Percentage CD8+ naive T cells

Percentage cells in division

Populations change by immigration, birth, and death processes, which could all depend on the density of the population itself

dN dt = [bf(N) − dg(N)]N

Population density Population density Per capita birth or death rate (a) (b)

death rate death rate birth rate birth rate steady state steady state

dN dt > 0 dN dt > 0 dN dt < 0 dN dt < 0

dN dt = h b − d ⇣ 1 + N k ⌘i N

d b b d

¯ N = k b − d d = k(R0 − 1)

dN dt = [b − d f(N)]N

Population density

F(N)

F(N) = d + cN = d f(N) ↔ f(N) = 1 + N/k

Population density

total death rate production rate steady state

dN dt > 0 dN dt < 0

Production or death rate

dN dt = s − d ⇣ 1 + N k ⌘ N

¯ N = −dk ± p dk(dk + 4s) 2d

s

Population density Population density Per capita birth or death rate (a) (b)

death rate death rate birth rate birth rate steady state steady state

dN dt > 0 dN dt > 0 dN dt < 0 dN dt < 0

d b b d

dN dt = [bf(N) − d]N

¯ N = k ⇣ 1 − d b ⌘ = k ⇣ 1 − 1 R0 ⌘

dN dt = h b ⇣ 1 − N k ⌘ − d i N

Population density

F(N)

F(N) = b − cN = bf(N) ↔ f(N) = 1 − N/k

dN dt = s ⇣ 1 − N k ⌘ − dN

Population density

total death rate production rate steady state

dN dt > 0 dN dt < 0

Production or death rate

¯ N = sk dk + s

s

(a)

Time N(t) K

(b)

N r(1 N/K) r K

(c)

N r(1 (N/K)m) K

Logistic growth: dN

dt = rN(1 N/K) , with solution N(t) = KN(0) N(0) + e−rt(K N(0))

m=0.5 m=2

(a)

Time N(t) K

(b)

N r(1 N/K) r K

(c)

N r(1 (N/K)m) K

Generalized logistic growth:

m=0.5 m=2

dN dt = rN(1 − (N/K)m) , with N(t) = K [1 − (1 − [K/N(0)]m)e−rmt]1/m

Human logistic growth

Human population in Monroe Country, West Virginia

(a)

Time N(t)

(b)

Time N(t)

dN dt = s ⇣ 1 − N k ⌘ − dN

dN dt = s − d ⇣ 1 + N k ⌘ N

Density dependent birth

Corn Salicornia Grizzly bear

10 100 100 1,000 10,000 Seeds planted per m2 Average number of seeds per reproducing individual (a) 10 20 30 40 Density of females 50 60 70 80 4.0 3.8 3.6 3.4 3.2 3.0 2.8 Clutch size Plantain (b) Song sparrow

Non-linear density dependence

f(x) = max(0, 1 − [x/k]n)

f(x) = min(1, [x/k]n)

f(x) = xn hn + xn

g(x) = 1 1 + (x/h)n

f(x) = 1 − e− ln[2]x/h

g(x) = e− ln[2]x/h

dN dt = (bf(N) − d)N ,

(a)

N f(N) k b d + −

(b)

N f(N) b k d + −

(c)

N f(N) b k d + −

f(N) = 1 1 + N/k , f(N) = 1 1 + [N/k]2 and f(N) = e− ln[2]N/k

dN dt = (bf(N) − d)N ,

(a)

N f(N) k b d + −

(b)

N f(N) b k d + −

(c)

N f(N) b k d + −

f(N) = 1 1 + N/k , f(N) = 1 1 + [N/k]2 and f(N) = e− ln[2]N/k

Function f(0) f(k) f(∞) R0 Carrying capacity Eq. f(N) = max(0, 1 − [N/k]m) 1 b/d ¯ N = k m p 1 − 1/R0 (3.12) f(N) = 1/(1 + N/k) 1 0.5 b/d ¯ N = k(R0 − 1) (3.14) f(N) = 1/(1 + [N/k]2) 1 0.5 b/d ¯ N = k√R0 − 1 (3.14) f(N) = e− ln[2]N/k 1 0.5 b/d ¯ N = (k/ ln[2]) ln[R0] (3.14)

Density dependent death

(a)

N d + δf(N) d d + δ

(b)

N d[1 + (N/k)m] d

dN dt = (b − d[1 + (N/k)m])N

dN dt = [b − d − δf(N)]N ,

• +

+

Positive density dependence

(a)

Time N(t) N(0) K

(b)

N bf(N)g(N)

dN dt = (bf(N)g(N) d)N

• r

dN dt = (bg(N) d[1 + (N/k)m])N ,

• +

Regression to the mean

n <- 100; data <- rnorm(n,1,0.1);hist(data) N <- data[1:(n-1)]; r <- (data[2:n]-N)/N plot(N,r,type="p") lm(r~N,as.data.frame(cbind(N,r)))

Smith-Martin model (first ignoring death):

is dN/dt = (p − d)N, w Cell division takes time

Conventional ODE:

dA(t) dt = 2pAt−∆ − pA(t) and dB(t) dt = pA(t) − pAt−∆

dA(t) dt = 2pAt−∆e−d∆−(p+d)A(t) and dB(t) dt = pA(t)−dB(t)−pAt−∆e−d∆

Smith-Martin model with death:

dA dt = 2n ∆ Bn−(p+d)A , dB1 dt = pA− ⇣ d+ n ∆ ⌘ B1 and dBi dt = n ∆ (Bi−1−Bi)−dBi

Time delays implemented as many small steps

Smooth the time delay by many (n) small steps:

dA(t) dt = 2pAt−∆e−d∆−(p+d)A(t) and dB(t) dt = pA(t)−dB(t)−pAt−∆e−d∆

Smith-Martin model with death: