Population models
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Population Models. What is a population? Populations are dynamic What factors directly impact dynamics Birth, death, immigration and emigration in models we frequently simplify things in order to gain a better understanding of how the rest will work E.g. a closed vs. open population.

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Population Models

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Population models

Population Models

  • What is a population?

  • Populations are dynamic

  • What factors directly impact dynamics

    • Birth, death, immigration and emigration

  • in models we frequently simplify things in order to gain a better understanding of how the rest will work

    • E.g. a closed vs. open population


Population models1

Population Models

  • Start with treating time as a ‘discrete’ (geometric population growth) unit rather than continuous (exponential growth)

  • Is this realistic? Why or why not?


Population models2

Population Models

Nt = Bt –Dt + It –Et

Nt+1 = Nt + Bt -Dt

  • Model development

  • Consider using per capita rates (individuals)

  • Rewrite the equation in terms of per capita rates:

  • With constant rates

bt = Bt/Nt anddt = Dt/Nt

Nt+1 = Nt + btNt - dtNt

Nt+1 = Nt + bNt - dNt


Population models3

Population Models

  • Model is somewhat realistic, but still useful

    • 1) provides a good starting point for more complex models (changes rates)

    • 2) it is a good heuristic – provides insight and learning despite its lack of realism

    • 3) many populations do grown as predicted by such a simple model (for a limited period of time)


Population models4

Population Models

  • Because this model does NOT change with population size, it is called density-independent

  • Furthermore, (b-d) is extremely important

  • λ is the finite rate of increase

Nt+1 = Nt + (b – d)Nt

Nt+1 = Nt + RNt

Nt+1 = (1+R)Nt

Nt+1 = λNt


Population models5

Population Models

  • Doubling time

  • Consider R=0.1; Λ=1+R (1.1)

Nt+1 = λNt

Nt double = 2N0

2N0 = λt double N0

Divide both sides by N0 : 2 = λt double

Take the logarithm of both sides: ln2 = tdouble lnλ

Divide both sides by lnλ: ln2 / lnλ = tdouble

7.27 years


Population models exponential growth continuous

Population Modelsexponential growth (continuous)

  • Instantaneous rate of change

  • Calculate the per capita rate of pop growth

  • Calculate the size of the pop at any time

dN / dt = rN

(dN / dt) / N = r

Nt = N0ert


Population models exponential growth continuous1

Population Modelsexponential growth (continuous)

  • Doubling Time

Nt double = N0ert double

Substitute 2N0 2N0 = N0ert double

Divide by N0 2 = ert double

Take natural log ln 2 = rtdouble

Finally divide by r tdouble = ln2 / r


Logistic population models

Logistic Population Models


Logistic population models1

Logistic Population Models


Logistic population models2

Logistic Population Models

  • Similarly this population model will explicitly model birth and death rates

  • Will also add in the concept of a carrying capacity (K), and one that is a continuous-time version


Logistic population models3

Logistic Population Models

  • Remember, the geometric model looked like this:

  • We can add two new terms to the model to represent changes in per capita rates of birth and death, where b’ and d’ = the amount by which the per capita birth or death rate changes in response to the addition of one individual of the pop(n)

Nt+1 = Nt + bNt - dNt

Nt+1 = Nt + (b+b’Nt)Nt – (d+d’Nt)Nt


Logistic population models4

Logistic Population Models

  • All four parameters (b, b’, d, d’) are assumed to remain constant through time (hence no bt)

  • How and why should b and d vary with density?

  • Logistic population models can be used to examine the potential impact of interspecificand intraspecificcompetition, as well as predator-prey relationships and harvesting populations


Logistic population models5

Logistic Population Models

  • We will explore the behavior of populations as numbers change

  • There is an equilibrium population size

Neq = b-d

d’-b’


Logistic population models6

Logistic Population Models

  • However, is it realistic to think populations will grow exponentially continuously?


Logistic population models7

Logistic Population Models

  • This equilibrium defined is so important, it is called the ‘carrying capacity’

  • This model gives us rate of change of population size

dN = rN {(K-N) /K)}


Logistic population models8

Logistic Population Models

  • To derive the equation for population size requires us to use calculus

Nt = K/ 1+ [(K-N0) / N0]e-rt


Logistic population models9

Logistic Population Models


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