class: left, top, inverse, title-slide .title[ # Interactions Part I: <strong>Competition and Coexistence</strong> ] .subtitle[ ## <a href="https://eligurarie.github.io/EFB370/">EFB 370: Population Ecology</a> ] .author[ ### <strong>Dr. Elie Gurarie</strong> ] .date[ ### March 27, 2024 ] --- ## Competition An **interaction** between organisms (*intraspecific*) or between species (*interspecific*) in which **fitness** of one is lowered by the presence of another. ***We've already talked about intraspecific competition!*** ### Fitness is Reproductive Success - survive to reproductive age - find a mate - produce (raise) offspring --- ## Competitive Exclusion Principle If two species are using the same resource and share the same niche, they can't coexist: .pull-left-70[] .pull-right-30[ Georgiy Franzevich Gause (1910-1986) ] --- ## Competitive Exclusion Principle .pull-left[  ] .pull-right[  across southern Minnesota prairie / farmland ] .center[[Levi and Wilmers (2021) *Ecology* 93(4)](https://doi.org/10.1890/11-0165.1)] --- ## But ... coexistence!? .pull-left-40[  ] .pull-right.large[ ### MacArthur's warblers Five species .... all insectivorous, all share the same set of boreal conifers in homogeneous stands. In principle (*specifically - according to the* ***competitive exclusion principle***), no 2 species should co-exist. .red[***How can 5 coexist!?***] ] --- ## Because **resource partitioning!** .center[ <img src='images/morewarblers.jpg' width='60%'/> ] --- ## Spatial niche partitioning .... .pull-left-40[] .pull-right-40[]  --- ## Temporal niche partitioning .... .pull-left[   ] .pull-right[  ] --- .pull-left-30[ ## Temporal niche partitioning LaPoint and Gurarie (in progress) ] .pull-right-70[] .content-box[ .pull-left-40[] .pull-right[] ] --- ## Apparent competition .large[Species A eats Species B and C, if Species B increases, Species C is in trouble.] .pull-left-60[] .pull-right-40[ Major habitat fragmentation from oil-gas extraction.  ***Serrouya et al. (2017)*** ] --- background-image: url("bg.jpg") background-size: cover # What can models tell us about coexistence and competition? --- ## Remember Logistic Growth Two species, `\(N_1\)` and `\(N_2\)`. .darkorange[ `$$\Large {dN_1 \over dt} = r_1 N_1 \left(1-{N_1 \over K_1}\right)$$` ] .darkblue[ `$$\Large {dN_2 \over dt} = r_2 N_2 \left(1-{N_2 \over K_2}\right)$$` ] each with its own `\(r\)` and `\(K\)`. --- ## Competition .... .darkorange[ `$$\large {dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)$$` ] .darkblue[ `$$\large {dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)$$` ] .large[ - `\(\alpha\)` - effect of `\(M\)` on `\(N\)` ... i.e. how many `\(N_1\)`'s does a single `\(N_2\)` cut things down by. - `\(\beta\)` - effect of `\(N\)` on `\(M\)`. ] --- ## Squirlicorn vs. Pegamunk Limited space | Limited carrying capacity | Mutual animosity (periodic horn skewering and/or dropping on rocks) .... .center[***Can they get along!?***]  https://egurarie.shinyapps.io/SquirlicornVsPegamunk --- ## Numerical Experiment #1 ### Q. How does initial population affect Co-Existence? .pull-left-60[ .box-green[ 1. Get in groups of 4. 2. Open the app. Set both **carrying capacities (K)** to 140. Leave the other parameters the same 3. Pick one person to be .red[Squirlicorn population], another to be .blue[Pegamunk population]. 4. Each of the **initial populations** ( `\(N_0\)` ) is your birth month x 10. 5. Run the experiment to time-step 100. Record the final number of .red[Squilicorns] and .blue[Pegamunks]. 6. Repeat 10 times, record the **average**, and enter data at front of room. 7. Second pair performs the same experiment. ] ] .pull-right-30[  ] --- ## Numerical Experiment #2 ### Q. How do different competition coefficients affect co-existence? .pull-left-60[ .box-green[ 3. Same setup: `\(K = 140\)`, but `\(N_0 = 40\)` for both species. 4. `\(\alpha\)` is a the .red[Squirlicorn's] meanness = date of birth divided by 30. 4. `\(\beta\)` is a the .blue[Pegamunk's] meanness = date of birth divided by 30. 5. Run the experiment to time-step 100. Record the final number of .red[Squilicorns] and .blue[Pegamunks] at the end, and if the process ended in coexistence. 6. Repeat 10 times, record the **averages**, and enter data at front of room. 7. Second pair performs the same experiment. ] ] .pull-right-30[  ] --- .pull-left[ ## Consider Species 1 The **special case** where there IS NO CHANGE in Species 1 population size: `$$0 = r_1 N_1 \left(1 - {N_1 \over K_1} - {\alpha N_2 \over K_1}\right)$$` boils down (quickly) to `$$\large N_1 = K_1 - \alpha N_2$$` ] .pull-right[ ### Isocline of Zero Growth: .darkorange[$$\Large {dN_1 \over dt} = 0$$] <img src="Interactions_PartI_Competition_files/figure-html/ZGI_N1-1.png" width="100%" /> ] --- ## How does a zero-growth isocline work? .pull-left-60[ <img src="Interactions_PartI_Competition_files/figure-html/ZGI_N1b-1.png" width="100%" /> ] .pull-right-40.large[ - Above the line, `\(N_1\)` decreases - Below the line, `\(N_1\)` increases ] --- .pull-left-40[ ## Add Species 2 Similarly assume .darkblue[Species 2] population is not changing. `$$0 = r_2 N_2 \left(1 - {N_2 \over K_2} - {\beta N_1 \over K_2}\right)$$` `$$N_2 = K_2 - \beta N_1$$` ] .pull-right-60[ ### Isocline of Zero Growth .darkblue[$$\Large {dN_2 \over dt} = 0$$] <img src="Interactions_PartI_Competition_files/figure-html/ZGI_parallel2-1.png" width="100%" /> ] --- .pull-left-30[ ## Follow the arrows .... ### To see species 1 drive species 2 to extinction: ] .pull-right-70[  ] --- .pull-left-30[ ## The lines don't have to be parallel ] .pull-right-70[  ] --- .pull-left-30.large[ ## What about a criss-cross? If `\(K_2 > {K_1 \over \alpha}\)` and `\(K_1 > {K_2 \over \beta}\)` *either* Species 1 or Species 2 will go **extinct**. ] .pull-right-70[  ] --- .pull-left-30.large[ ## What about a criss-cross? BUT ... If `\(K_2 < {K_1 \over \alpha}\)` and `\(K_1 < {K_2 \over \beta}\)` There will be .purple[**coexistence!**] At an equilibrium that is (of course) smaller than either `\(K_1\)` or `\(K_2\)`. ] .pull-right-70[  ] --- class: inverse ## Competition .... .orange[ `$$\Large N_2 = {K_1 \over \alpha} - {1\over \alpha} N_1$$` ] .cyan[ `$$\Large N_2 = K_2 - \beta N_1$$` ] .large[ This model captures TWO density dependent phenomena: 1. Intra-specific competition ( `\(K_1\)` and `\(K_2\)` ) 2. Inter-specific competition ( `\(\alpha\)` and `\(\beta\)` 2) Patterns depend on the *scale* and *particulars* of the competition intensity. ] --- .pull-left-40[ ## Focus on the the isoclines of zero growth .darkorange[ `$$\Large N_2 = {K_1 \over \alpha} - {1\over \alpha} N_1$$` ] .darkblue[ `$$\Large N_2 = K_2 - \beta N_1$$` ] ] .pull-right-60[ <!-- --> ] --- .pull-left-40[ ## Extinction vs. Equilibria If: - `\({1\over\beta} < {K_1 \over K_2} > \alpha\)`; .darkorange[ N₁ ] wins - `\({1\over\beta} > {K_1 \over K_2} < \alpha\)`; .blue[ N₂ ] wins - `\({1\over\beta} < {K_1 \over K_2} < \alpha\)`; .darkorange[ N₁ ] or .blue[ N₂ ] wins - `\({1\over\beta} > {K_1 \over K_2} > \alpha\)`; .darkorange[ N₁ ] and .blue[ N₂ ] coexist Equilibria (stable or not) at: `$$N^*_1 = {K_1 - \alpha K_2 \over 1-\alpha\beta}$$` `$$N^*_2 = {K_2 - \beta K_1 \over 1-\alpha\beta}$$` ] .pull-right-60[ <!-- --> ] --- ## Co-existence essentially occurs when .... .pull-left.large[ **Intra**-specific > **Inter**-specific competition `$${1\over\beta} > {K_1 \over K_2} > \alpha$$` i.e. when `\(\beta\)` and `\(\alpha\)` are small (in a very particular way) relative to the ratio of carrying capacities. ### These is known as **"Lotka-Volterra competition model"** ] .pull-right[  ] --- background-image: url('images/LotkaVolterra.jpg') background-size: cover .pull-left[ ### Vito Volterra (1860-1940) Italian mathematician, father-in-law of fisheries biologist. One of 12 (out of 1250) of Italian professors to not sign an oath of loyalty to Mussolini. ] .pull-right[ ### Alfred J. Lotka (1880-1949) Chief statisticitan at Met Life Insurance. And that's almost all there is to know. ] .pull-right-70[.pull-left-50[ > Independently published these equations, with more or less this exact application, in 1926 and 1925. ]] --- ## Nice theory you've got there gentlemen, but ..... .pull-left-40[ ... can you detect or quantifying competition in anything resembling a real system!? ] .pull-right-60[] --- background-image: url('images/gerbils.png') background-size: cover --- ## Gerbil coexistence and habitat partitioning .pull-left-30[  ] .pull-right-70[ - **Zone I**, both species prefer the semi-stabilized dune. - **Zone II** *G. pyramidum* prefers the semi-stabilized dune and *G. allenbyi* exhibits random utilization of both habitats - **Zone III** *G. allenbyi* exhibits preference for the stabilized sand and *G. pyramidum* prefers semi-stabilized dune. - **Zone IV** *G. allenbyi* prefers stabilized sand and *G. pyramidum* selects semi-stabilized dune but also uses the stabilized sand. ] --- ## Ultimately Gerbils **CAN** get along. .... .pull-left[ ### But the isoclines are complicated!  ] .pull-right-40[  .small[image: SamWildflower] ]