The first law of populations: Population Growth The first law of populations: “A population will grow (or decline) exponentially as long as the environment experienced by all individuals remains constant” -P. Turchin. 2001. Oikos 94:17-26.

Darwin’s elephants “…after a period of 740 to 750 years there would be nearly 19 million elephants alive, descended from the first pair.”

Any population may show this Is this realistic? Actually, yes. Any population may show this growth for short periods of time Asking why it ISN’T happening is extremely valuable

Population growth Nt+1 = Nt + B + I - D- E Let’s assume geographic closure: Nt+1 = Nt + B - D Let’s also assume continuous population growth….

Continuous population growth dN/dt = B - D “the change in population size with respect to time is due to the difference between number of births and deaths” b = per capita birth rate d = per capita mortality rate dN/dt = (b - d)N

The exponential growth rate: r Let (b- d) = r; The equation becomes: dN/dt = rN The change in population size with respect to time is a function of the exponential growth rate multiplied by the population size

Can we predict population size into the future? dN/dt = rN Using calculus, we can integrate this Into the equation, Nt = N0ert This is known as the exponential equation It describes continuous population growth

Modeling grizzly bears in Yellowstone Gotelli 2001 p. 7

BUT… grizzlies don’t have continuous reproduction… We can use a discrete-time model to represent breeding seasons separated by non-breeding seasons N(t+1) = N(t) + bN(t) – dN(t) N(t) = λtN(0) where λ = 1 + R or, more generally,

The trajectory might look like this… Gotelli 2001 p. 12

Some terminology r = exponential growth rate λ = geometric growth rate

Relationship between r and λ er = λ r = ln(λ) or If r is … then λ is….and the population is 0 1 stable <0 <1 decreasing >0 >1 increasing

How do exponential and geometric growth differ? The implications of continuous versus discrete reproductive periods….

Some assumptions about exponential and geometric growth Geographic closure individuals added only by birth, lost only by death Birth and death rates are constant This means growth rate is constant No age/stage structure No time lags

Constant growth rate?? This means (b – d) must be constant, and it also implies that resources are unlimited Real populations are like this for short periods of time This idea is the key to natural selection and evolution Clearly exists for insect pest outbreaks, invasive exotic species, Homo sapiens

That “no structure” assumption… (just what IS structure, anyway??) Can we reasonably expect a population to have neither age nor stage structure? Sometimes, yes. What if we can’t? How does it matter?

Consider the difference… Which one has more growth potential? Why?

Consider the difference… 2 yearlings 10 yearlings 3 first-year 5 first-year 15 2-year 5 2-year Which one has more growth potential? Why?

Structure and population growth If individuals differ in reproductive rates or survival rates “enough” and these differences can be summarized by age or stage, we can project population growth using a structured model.

A stage-classified model: An example… A simplified model for Strix Occidentalis First breeding: 2 years Maximum age: ~ 15 years A stage-classified model: J A

Unstructured versus stage-classified model No structure: N(t+1) = λ * N(t) Growth rate is the difference between births and deaths, or number surviving plus the number of their offspring that also survive… N(t+1) = (S + F) * N(t) N(t+1) = N(t) * S + N(t) * F

Unstructured versus stage-classified model Structured model: Survival and reproduction Are stage-specific Add Structure… N(t+1) = N(t) * S + N(t) * F N(t+1) = N(t)adults * Sadults + N(t)juveniles * Sjuveniles + N(t)adults * fertility S F Fertility: the number of young per adult surviving to enter the population

We can rewrite this: N(t+1)juveniles = N(t)adults * F N(t+1)adults = N(t)juveniles * Sj + N(t)adults * Sa ______ N(t+1)total owls

Unstructured versus stage-classified model Adding structure to our model lets us track number of individuals in each stage through time as well as total population numbers When do we need to add structure? -When demographic rates vary by age/stage -When we have the information on those demographic rates

Review Biological populations will grow at a constant rate unless prevented from doing so This constant growth can be modeled as either exponential or geometric Exponential growth: continuous time r: the exponential growth rate Geometric growth: discrete time steps λ: geometric growth rate

Correctly accounts for reproduction (time Step = breeding interval) Review NOTE CHANGE: Exponential and Geometric Models give same results if geometric model Correctly accounts for reproduction (time Step = breeding interval) Geometric and exponential growth assume: Constant demographic rates No structure Geographic closure (no I,E) No time lags

Review Populations differ in structure, and this affects their growth potential Structured models should be used when: -Demographic rates differ by age/stage -When the information is available

UNGRADED HOMEWORK 1). What is the difference between geometric And exponential population growth? 2). What is the first law of populations?