1. Invasive Species • Species occasionally invade new habitat, irrespective of whether intentional or unintentional result of human activities • If the population becomes successfully established, the population will grow and eventually need to expand its range

2. Invasive Species • Considering the potential negative impact of invasive species, it is important we better understand the how’s and why’s of species range expansion • What are the two factors that influence range expansion (or contraction) rate? Population Growth Emigration (or Dispersal) Rate

3. Invasive Species • In this exercise, you will treat each cell in the spreadsheet as a patch of habitat, which may house a local population • Each population may grow or shrink according to its birth and death rates and it may exchange members with neighboring local populations through immigration or emigration • The number of cells occupied by local populations is the range of the population

4. Invasive Species • We begin by assuming that the local population in each cell grows according to a logistic model: b’ and d’ represent per capita rates when populations are small and uncrowded • However, this model does not take into account any I or E Nt+1 = Nt + (b+ b’Nt)Nt – (d + d’Nt)Nt

5. Invasive Species • We could either add more terms to the model, or simply the model • If we insert Rmax to represent the population’s maximum geometric rate of growth and Rdd for the density-dependent reduction in population growth rate Nt+1 = Nt + (b+ b’Nt)Nt – (d + d’Nt)Nt Nt+1 = Nt + bNt – dNt + b’Nt2+ d’Nt2 Nt+1 = Nt + (b - d)Nt – (b’ - d’)Nt2

6. Invasive Species Nt+1 = Nt + (b - d)Nt + (b’ - d’)Nt2 • Let Rmax = b-d and Rdd = b’-d’ • This is the same as the logistic model earlier • Let’s add E Nt+1 = Nt + Rmax*Nt + Rdd*Nt2 Nt+1 = Nt + (Rmax* RddNt)Nt

7. Invasive Species Here Eminrepresent the minimum emigration rate and Edd the density-dependent increase in emigration Emigrants have to go somewhere and this model assumes equal likelihood to any adjacent cell Nt+1 = Nt + (Rmax* RddNt)Nt – (Emin + EddNt)Nt

8. Invasive Species • From the perspective of the ‘target’ cell, immigrants move in from neighboring cells at rates determined by the sizes of the populations in those neighboring cells (L & R) • The population of each cell grows by reproduction in its own cell and loses members by emigration. It does receive some members from L or R Nt+1 = Nt + (Rmax* RddNt)Nt – (Emin + EddNt)Nt + 0.5(Emin + EddLt)Lt + 0.5(Emin+EddRt)Rt

9. Invasive Species • This model can be adjusted to represent geometric population growth as well

10. Invasive Species • Objectives • Build a spatially explicit model of range expansion (or invasion) by a logistically growing population • Model the expansion of a species’ range in one and two dimensions • Determine how the rate of range expansion relates to population growth and emigration