FW 662 – Density-dependent population models

In the previous lecture we considered density independent population models that assumed that birth and death rates were constant and not a function of population size. Long-term density independent population growth is unlikely and an unrealistic assumption. Birth and death rates are more likely a function of population density or abundance.

Density Dependence births are a decreasing function of density b(N) and deaths are an increasing function of density d(N).

births deaths d b or

N

This results in population growth being a declining function of N

) N ( th (f w o r G on ti a l pu o P

N

Hence population growth will be zero at some population size. This point is usually referred to as K (or carrying capacity) but let’s develop the model first considering the explicit functions for birth and death. The approach is described in Donovan and Weldon (2002) but I have modified it to match the notation in Gotelli (1998).

FW 662 – Density-dependent population models

We need two new terms to account for changes in per capita birth and death rates

a= the amount by which the per capita birth rate changes in response to an addition of one individual to the population.

c= ditto for death rate…

Our density independent discrete model was a difference equation expressed as:

N t+1 = N t + bN t − dN t

We replace b and d (the density independent birth rate and death rate) with:

b − aN t and d + cN t

Now our new density dependent model looks like

N t+1 = N t + ()b − aN t N t − (d + cN t )N t

Population growth rate is not easy to visualize from this equation. But you can see that initially the population will grow geometrically because N(t) is small but as N(t) grows a and c have a greater influence on the population.

The question is will the population decrease in size? Increase? Or stabilize???

Use a common tactic and assume that there is an equilibrium (i.e. a point where the population size isn’t changing).

N t = N t+1 = N eq

N eq = N eq + (b − aN eq )N eq − (d + cN eq )N eq

Subtract N(eq) from both sides and add (d + cN eq )N eq

(b − aN eq )N eq = (d + cN eq )N eq

Divide by N(eq)

(b − aN eq ) = (d + cN eq )

This tells us that the population is at equilibrium when per capita birth and death rates are equal. THIS MAKES SENSE. FW 662 – Density-dependent population models

Now we can solve the equation for N(eq)

Rearrange the equation by subtracting d and adding aN

b − d = cN eq + aN eq

Factor out N(eq)

b − d = ()a + c N eq

Divide by (a+c)

b − d = N = K a + c eq

Using this equation you can calculate K, if you know b, d, and the two factors that account for density dependence.

Density Dependence (discrete logistic)

Typically, K is specified and not calculated from the birth and death rates.

Let’s start with our geometric model

∆N t = RN t

The existence of a carrying capacity (K) suggests that the population cannot exceed this level.

⎛ N t ⎞ ∆N t = RN t ⎜1− ⎟ ⎝ K ⎠

⎛ N t ⎞ N t+1 − N t = RN t ⎜1− ⎟ ⎝ K ⎠

⎛ N t ⎞ N t+1 = N t + RN t ⎜1− ⎟ ⎝ K ⎠

FW 662 – Density-dependent population models

The logistic model results in the following population dynamics. When N(t) is small then geometric growth and when N(t)=K then population growth is zero.

Pop Siz e

60

50 40

N 30 20

10 0 0 5 10 15 20 25 Time

We can also look at the change in population size as a function of population size. Initially the population is growing quickly but then decline to zero.

Delta N( t)

10

8

N 6 a t

Del 4

2

0 0204060 N

Looking at the per capita rate of change, the population begins growing at R and declines to zero at K. From this graph you can see that the density dependence is linear.

(DeltaN(t))/N(t)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0204060 N FW 662 – Density-dependent population models

Density Dependence (continuous logistic)

dN ⎛ K − N ⎞ = rN⎜ ⎟ dt ⎝ K ⎠

This equation predicts the rate of change in population size and you need to derive a predictive equation.

WHICH IS

K N t = ⎡()K − N 0 ⎤ −rt 1+ ⎢ ⎥e ⎣ N 0 ⎦

This model results in the same dynamics that were demonstrated for the discrete version of the model. However, the discrete version of the model is capable of demonstrating a wider variety of dynamics than the continuous version, which we will explore later.

We have derived two predictive equations for exponential population growth and two equations for density dependent population growth. Discrete Continuous rt Density Independent t N N e Nt = λ N0 t = 0 K N = K − N t ⎛ t ⎞ ⎡ ⎤ Density Dependent N t+1 = N t + RN t ⎜ ⎟ ()K − N0 −rt ⎝ K ⎠ 1+ ⎢ ⎥e ⎣ N0 ⎦

FW 662 – Density-dependent population models

Other models with Density Dependence

PLEASE PRINT AND READ GARY’S LECTURE NOTES (5-7)

Many possibilities exist for describing density dependence and there are several other models that have been developed.

The Ricker equation

Ricker was a fishery biologist interested in predicting recruitment to fishery stocks and developed the following density dependent equation. Note that density dependence in this equation is not linear and becomes stronger at higher densities (due to the exponential function).

⎛ Nt ⎞ R0 ⎜1− ⎟ ⎝ K ⎠ Nt +1 = Nte

The Hassel Equation

KN (R +1) N = t 0 t+1 K + Nt R0

FW 662 – Density-dependent population models

Comparing population growth among the logistic, Ricker, and Hassel models for R=0.75 and K=100.

120

100

80

Logistic (t) 60 N Ricker 40 Hassel

20

0 0 5 10 15 20 25 Time (t)

Comparing the form of density dependence among the logistic, Ricker, and Hassel models.

1.2 Logistic 1 )

t Ricker (

N 0.8 Hassel

(t) / 0.6 N

a t l 0.4 e d 0.2

0 0 20 40 60 80 100 120 N(t)

Comparing the per capita growth rate among the logistic, Ricker, and Hassel models

25 Logistic 20 Ricker Hassel

(t) 15 N a t l

e 10 d

5

0 0 20406080100120 N(t) FW 662 – Density-dependent population models

Maximum Sustainable Yield (MSY)

I will develop a model based on the logistic difference (discrete) equation and follow ther derivation in Williams et al (2002).

Start with our discrete logistic difference equation

⎛ N t ⎞ N t+1 = N t + RN t ⎜1− ⎟ ⎝ K ⎠ and include a term for total Harvest H(t)

⎛ N t ⎞ N t+1 = N t + RN t ⎜1− ⎟ − H t ⎝ K ⎠

The population is at equilibrium (N(t+1)=N(t)) when

⎛ N t ⎞ RN t ⎜1− ⎟ = H t ⎝ K ⎠

and the per capita harvest rate h

H t ht = N t

⎛ N t ⎞ ht = R⎜1− ⎟ ⎝ K ⎠

A given per capita harvest rate h(t) corresponds to a specific equilibrium population size that can sustain it and can be seen by rearranging the above equation in terms of N.

⎛ h ⎞ N = K⎜1− ⎟ ⎝ R ⎠

This equation indicates that the population can be sustained in equilibrium for any value h that is less that the population growth rate R. The question is, “What value corresponds to the largest sustainable harvest?”

dH = R − 2R()N dN K

FW 662 – Density-dependent population models

Set this equal to 0 and solve for N, which results in

N ∗ = K 2

Substituting into our equation for equilibrium harvest:

∗ ∗ ⎛ N ⎞ ∗ RN ⎜1− ⎟ = H ⎝ K ⎠

R N ∗ = H ∗ ( 2)

RK = H ∗ ( 4)

Then the per capita harvest (h) is

∗ h∗ = H N ∗

h∗ = R 2

FW 662 – Density-dependent population models

MSY 14 12 )

t 10

N( 8 a t 6

Del 4 2 0 0 100 200 300 400 500 N(t )

MSY 0.11

) 0.1 t

( 0.09

N 0.08 0.07 t))/

( 0.06

N 0.05

ta 0.04 l

e 0.03 0.02 (d 0.01 0 0 100 200 300 400 500 N(t)

FW 662 – Density-dependent population models

Gotelli, N. 1998. A primer of ecology. Sinauer Associates Inc. New York.

Hassel, M.P. Density-dependence in single-species populations. The Journal of Animal Ecology. 44:283-295.

Ricker, W. E. 1954. Stock and recruitment. Journal Fisheries Research Board of Canada 11:624-651.

Ricker, W. E. 1975. Computation and interpretation of biological statistics of fish populations. Fisheries Research Board of Canada, Bulletin 191. Ottawa, Canada.

Williams, B.K., J.D. Nichols, M.J. Conroy. 2002. Analysis and management of animal populations. Academic Press. New York.