Ecology Exam 2 Equations

N = nM / m

Mark-Recapture

N = whole population

M = individuals captured and marked the first time

n = total individuals in second capture

m = individuals that were marked from the second recapture

N_t+1 = N_t𝛌

Discrete-time population growth model

N = population abundance

𝛌 = 1+b-d

b = per capita birth rate

d = per capita death rate

N_t = N₀𝛌^t

Geometric population growth model (still discrete)

N₀ = original population size

Also 𝛌 = (N_t/N₀)^1/t

N_t = N₀e^rt

Continuous time population growth model

r = instantaneous per capita rate of population growth (b_{instantaneous} - d_{instantaneous})

N₀ = original population size

t = time

e = mathematical placeholder, can be subbed as π

dN/dt = Nr_max (K-N / K)

Continuous-time logistic growth model (density dependent)

K = carrying capacity

N = population abundance

r_max = intrinsic rate of increase (maximum per capita growth)

dN/dt = rate of change of population over time

(K-N/K) = fraction of carrying capacity remaining at population size N

N_t+1 = N_t + N_tr_dis (K-N_t / K)

Discrete-time density-dependent logistic growth model

r_dis = maximum per capita rate of population change

N_t+1 = population density at the next time step

N_t = current population density

K = carrying capacity

dN₁/dt = r₁N₁ (K₁-N₁-𝜶N₂ / K₁)

and

dN₂/dt = r₂N₂ (K₂-N₂-𝜷N₁ / K₂)

Lotka-volterra models for competition

𝜶N₂ and 𝜷N₁ are competition coefficients that link the two species (conversion factor)

r = maximum per capita growth rate for each population

N = population density for each population

K = carrying capacity

dN_p/dt = rN_p - fN_pN_e

Lotka-Volterra prey equation

N_p = prey population density

N_e = exploiter population density

r = intrinsic rate of increase in prey population

f = capture rate (success and eating mode)

dN_e/dt = cfN_pN_e - dN_e

Lotka-volterra exploiter equation

N_p = prey population density

N_e = exploiter population density

d = per capita death rate of exploiter

f = capture rate

c = exploiter conversion factor (consumed food to expected offspring)

dN_p/dt = r_maxN_p (K-N_p / K) - fN_pN_e

Lotka-volterra prey equation with density dependence

(K-N_p / K) is added to the prey equation

N_p = prey population density

N_e = exploiter population density

r = intrinsic rate of increase in prey population

f = capture rate (success and eating mode)

K = prey carrying capacity