Population Growth Models: Geometric Growth Brook Milligan - - PowerPoint PPT Presentation

Population Growth Models: Geometric Growth

Brook Milligan

Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu

Fall 2009

Brook Milligan Population Growth Models: Geometric Growth

Population Models in General

Purpose of population models Project into the future the current demography (e.g., survivorship and reproduction) Guage the potential (or lack) for a population to increase Determine the consequences of changes in the current demography

Brook Milligan Population Growth Models: Geometric Growth

Population Models in General

Observables: N or N(age) or N(stage) Project population size N as a function of time t Projection in terms of fundamental parameters

describing demographic events in an individual’s life e.g., Pr(birth), Pr(death)

enable understanding of how demographic vital rates affect the whole population

Brook Milligan Population Growth Models: Geometric Growth

Projection versus Prediction

No population experiences unlimited resources Yet, all populations have potential for exponential growth Projections describe potential, not what is actually predicted to occur

analogy: a speedometer projects potential travel only

Brook Milligan Population Growth Models: Geometric Growth

Geometric Growth Models

General motivation Sequence of population sizes through time Nt, Nt+1, Nt+2, . . . Change from one time to next

increases due to births during period decreases due to deaths during period increases due to immigrants during period decreases due to emigrants during period

Brook Milligan Population Growth Models: Geometric Growth

Mathematical Formulation

Population size after an interval of time Nt+1 = Nt + B − D + I − E (1)

B, D: birth, death I, E: immigration, emigration

Change in population size ∆N = Nt+1 − Nt (2) = B − D + I − E (3) Closed versus open populations

Brook Milligan Population Growth Models: Geometric Growth

Geometric Growth Model: Assumptions

Closed population: I = E = 0 Constant per captita birth (b) and death (d) rates

B = bN D = dN

Brook Milligan Population Growth Models: Geometric Growth

Geometric Growth Model: Assumptions

Closed population: I = E = 0 Constant per captita birth (b) and death (d) rates

B = bN D = dN

Unlimited resources No genetic structure

b and d identical for all individuals regardless of genotype

No age- or size-structure

b and d identical for all individuals regardless of size, age, . . .

No time lags

birth and death depend on current population only

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Growth: Projection of Population Size

Nt+1 = Nt + B − D (4) = Nt + B Nt − D Nt

• · Nt

(5) = Nt + (b − d) · Nt (6) = (1 + (b − d)) · Nt (7) = (1 + Rt) · Nt (8) = λt · Nt (9) λt = Nt+1 Nt (10)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Growth: Change in Population Size

∆N = Nt+1 − Nt (11) = (1 + Rt)Nt − Nt (12) = Nt + RtNt − Nt (13) = RtNt (14) Rt = ∆N Nt (15)

Brook Milligan Population Growth Models: Geometric Growth

Finite Rate of Increase: λ

Nt+1 = λtNt (16) λt = Nt+1 Nt (17) population increase: λ > 1 population stable: λ = 1 population decrease: λ < 1

Brook Milligan Population Growth Models: Geometric Growth

Projection of Population Size

Assume a constant value of λ: i.e., λt = λ N1 = λN0 (18) N2 = λN1 (19) = λ(λN0) (20) = λ2N0 (21) Nt = λNt−1 (22) = λ(λNt−2) (23) = λ(λ(λNt−3)) (24) = λtN0 (25)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Doubling Time

How long does it take the population to double in size? That is, how long does it take the population to change from N0 to 2N0? Nt = λtN0 (26) 2N0 = λtN0 (27)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Doubling Time

How long does it take the population to double in size? That is, how long does it take the population to change from N0 to 2N0? Nt = λtN0 (26) 2N0 = λtN0 (27) 2 = λt (28)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Doubling Time

How long does it take the population to double in size? That is, how long does it take the population to change from N0 to 2N0? Nt = λtN0 (26) 2N0 = λtN0 (27) 2 = λt (28) ln(2) = ln(λt) (29) ln(2) = t · ln(λ) (30) t = ln(2) ln(λ) (31)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Half Life

How long does it take the population to become half as large in size? That is, how long does it take the population to change from N0 to 1

2N0?

Nt = λtN0 (32) 1 2N0 = λtN0 (33)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Half Life

How long does it take the population to become half as large in size? That is, how long does it take the population to change from N0 to 1

2N0?

Nt = λtN0 (32) 1 2N0 = λtN0 (33) 1 2 = λt (34)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Half Life

How long does it take the population to become half as large in size? That is, how long does it take the population to change from N0 to 1

2N0?

Nt = λtN0 (32) 1 2N0 = λtN0 (33) 1 2 = λt (34) ln(1 2) = ln(λt) (35)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model: Half Life

How long does it take the population to become half as large in size? That is, how long does it take the population to change from N0 to 1

2N0?

Nt = λtN0 (32) 1 2N0 = λtN0 (33) 1 2 = λt (34) ln(1 2) = ln(λt) (35) − ln(2) = t · ln(λ) (36) t = − ln(2) ln(λ) (37)

Brook Milligan Population Growth Models: Geometric Growth

Geometric Population Model

Quantitative description of how a population changes size as time progresses Depends directly on the finite rate of increase, λ λ in turn depends on the per capita rates of birth and death (through their difference only) λ measures the rate of increase λ measures the potential for a population to grow Questions that can be answered:

Is the population increasing, decreasing, or stable? What is the potential for the population to increase? How long does it take for the population to change by a certain amount? How will the answers change if the vital rates (b and d) change?

Brook Milligan Population Growth Models: Geometric Growth