class: center, middle, inverse, title-slide # Lecture 15 ## Metapopulation dynamics II ### <br/><br/><br/>WILD3810 (Spring 2020) --- ## Reading > ### Mills 188-196 --- ## Assumptions of the Levins model - `\(\large \gamma\)` and `\(\large \epsilon\)` are the same for every patch <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are constant over time <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are independent of patch size <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are independent of distance of patch to other patches <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are independent of population density <br/> - Local birth-death dynamics are ignored --- ## Assumptions of the Levins model - `\(\large \mathbf \gamma\)` **and** `\(\large \mathbf \epsilon\)` **are the same for every patch** <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are constant over time <br/> - `\(\large \mathbf \gamma\)` **and** `\(\large \mathbf \epsilon\)` **are independent of patch size** <br/> - `\(\large \gamma\)` **and** `\(\large \epsilon\)` **are independent of distance of patch to other patches** <br/> - `\(\large \gamma\)` and `\(\large \epsilon\)` are independent of population density <br/> - Local birth-death dynamics are ignored --- ## Island biogeography #### About the same time Levins was developing the concept of Metapopulation dynamics, Robert MacArthur and E.O. Wilson were developing their theory of Island Biogeography (1963, 1967) <img src="Lecture15_files/figure-html/unnamed-chunk-1-1.png" style="display: block; margin: auto;" /> --- ## Island biogeography <img src="Lecture15_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> - **Immigration curve**: successful colonization decreases as number of species increases because niches and habitats become filled - **Extinction curve**: competition increases with increasing number of species, leads to higher extinction rate --- ## Island biogeography <img src="Lecture15_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> `$$\LARGE \Delta S = Imm - Ext$$` --- ## Island biogeography <img src="Lecture15_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" /> `$$\LARGE \Delta S = Imm - Ext$$` --- ## Island biogeography <img src="figs/mainland.png" width="35%" style="display: block; margin: auto;" /> --- ## Island biogeography #### Immigration curves should be lower for **far** islands than for **near** ones: - distant islands are less likely to be colonized <img src="Lecture15_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" /> --- ## Island biogeography #### Immigration curves should be lower for **small** islands than **large** ones: - smaller islands offer a smaller target to find <img src="Lecture15_files/figure-html/unnamed-chunk-7-1.png" style="display: block; margin: auto;" /> --- ## Island biogeography #### Extinction curves should be higher for **small** islands than for **large** ones: - Small islands support smaller population sizes; increased risk of extinction via demographic stochasticity <img src="Lecture15_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" /> --- ## Island biogeography .pull-left[ <img src="figs/S_eq.png" width="100%" style="display: block; margin: auto;" /> ] .pull-left[ <img src="figs/mainland.png" width="70%" style="display: block; margin: auto;" /> ] --- ## Island biogeography #### How can the predictions of island biogeography be tested experimentally? -- - Wilson andhis grad student Daniel Simberloff de-faunated small clumps of mangroves in Florida bay that differed in their distance from the mainland .pull-left[ <img src="figs/mangrove.png" width="90%" style="display: block; margin: auto;" /> ] .pull-left[ <img src="figs/wilson_simberloff.png" width="80%" style="display: block; margin: auto;" /> ] --- ## Island biogeography - **Prediction 1**: insect diversity would attain pre-removal levels - **Prediction 2**: close islands would be colonized more quickly compared tomore distant islands .pull-left[ <img src="figs/mangrove.png" width="90%" style="display: block; margin: auto;" /> ] .pull-left[ <img src="figs/wilson_simberloff.png" width="80%" style="display: block; margin: auto;" /> ] --- ## Island biogeography <img src="figs/wilson_simberloff2.png" width="80%" style="display: block; margin: auto;" /> --- ## Island biogeography #### Island Biogeography led to concept of **Mainland-Island metapopulation structure** <img src="figs/mainland_source.png" width="50%" style="display: block; margin: auto;" /> --- ## Mainland-island #### Large, stable Mainland population: - Provides a constant supply of dispersing individuals - Colonization does not depend on fraction of islands occupied because colonization largely comes from the mainland `$$\LARGE \hat{\psi} = \frac{\gamma}{\gamma + \epsilon}$$` - Guarantees that at least some islands are occupied because of continual flood of immigration from mainland --- ## Implications of IB and M-I for habitat conservation #### Preserving single large refuge and tract of habitat - Supports greater local abundance - Has low chance of local extinction - Supports more species --- ## Implications of IB and M-I for habitat conservation #### Preserving single large refuge and tract of habitat - Supports greater local abundance - Has low chance of local extinction - Supports more species - Vulnerable to isolated catastrophe -- #### Preserving several small tracts of habitat - Higher chance of local extinctions - More robust to isolated catastrophes - Connectivity amongst habitat patches is crucial