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Competition • Different kinds of competition • Modeling competition • Examples of competition-case studies • Understanding the role of competition Competition • The outcome of competition is that an individual suffers a reduction in fecundity (ability to reproduce), growth or survivorship • Competition is for a limiting resource Intraspecific competition (between members of own species) Bald Eagles with disagreement over life mate Olympic ski cross Olympic short track skaters disagree on potential podium position Snails in deathmatch over habitat Interspecific competition (between members of different species) Eagle vs fox - fight for control of deer carcass Yoda vs Darth Vader -fight for control of coffee shop aboard the Death Star Types of competition Resource exploitation - indirect White pine roots are better able to take up moisture and nutrients compared to other plants. This prevents other plants from accessing the limited resources. White pine indirectly suppresses the other plants. fig 13.2 Molles and Cahill Humans have become super efficient at taking up fish from oceans and suppress other animals that depend on fish resources. Interference competition-direct competition Damsel Fish aggressively maintain territories. fig 13.1 Molles and Cahill Humans aggressively maintain territories. Interference competition ___________________________________________ Exploitative competition • Competition is NOT always the most important type of interaction between or amongst members of species • • • Abiotic (environmental) stress Predation Parasites May individually or in combination play a greater role in limiting fitness, population dynamics and community structure • Mathematical Modeling of Competition Why?? What is a model? • Simplification of a system (nature) • It is NOT a working facsimile! • Used to gain insight into how things work • Should be testable and modifiable “A model should be as simple as possible but not simpler! “ Albert Einstein What does this mean? Make the irreducible basic elements of your model as simple and as few as possible without having to give up the adequate representation of what you are modeling From last class. . . Molles 2008 (page 261) cf. Molles & Cahill, 2008, p. 316 dN =r N max dt ( K-N K ) ( ) ( ) dN =r N K1-N1 max 1 dt K1 dN =r N K2-N2 max 2 dt K2 Logistic model for population growth Change in population growth of Species 1 Change in population growth of Species 2 In these models population growth slows as N increases since resource supplies decrease with as population increases N1 K1 = Relative level of intraspecific competition But resource levels can also be reduced by interspecific competition Lotka Volterra Models of Interspecific Interations Lotka Volterra Models dN1 dt =rmax1 N1 dN2 dt =rmax2 N2 ( K - N K- α 1 12N2 1 1 ( K - N K- α 2 21N1 2 2 ) ) These equations are same as before but now include the effect of interspecific competition (in red). We now can simultaneously model the effect of individuals of the same species and individuals of competing species on the rate of growth of a population Lotka Volterra Models a12 and a21 are competition coefficients that modify the effect of N2 and N1 (remember N1 and N2 are population sizes) dN1 dt =rmax1 N1 ( K - N K- α dN2 =r dt max2 N2 1 12N2 1 1 ( K - N K- α 2 ) 21N1 2 2 ) α12 = effect of an individual of species 2 on the rate of population growth of species 1 α21= effect of an individual of species 1 on the rate of population growth of species 2 a12 < 1 means that the individuals of species 2 have less effect on individuals of species 1 than individuals of species 1 have on others of their own species. a12 >1 means that the individuals of species 2 have more effect on individuals of species 1 than individuals of species 1 have on others of their own species. Lotka Volterra Models Predicts: i) coexistence of two species when for both species interspecific competition is weaker than intraspecific competition ii)If the above condition is NOT met than one species will eventually exclude the other Lotka Volterra Models Can we determine under what conditions these species are predicted to coexist and under what conditions one species will exclude the other? To do this we determine equilibria: population sizes for species 1 and 2 for which population growth of both species will be zero.! If population growth is zero, then the population sizes do not change over time, and we have an equilibrium (a situation in which conditions remain the same over time.) dN1 dt =rmax1 N1 ( K - N K- α 12N2 )= 0 dN2 =r max2 N2 dt ( K - N K- α 21N1 )= 0 1 1 1 2 2 2 Populations stop growing when: dN1 dt =rmax1N1 ( K - N K- α dN2 dt =rmax2 N2 1 12N2 1 1 ( K - N K- α 2 )= 0 21N1 2 2 )= 0 This occurs when 0 = (K1 - N1 - α12N2) 0 =(K2 - N2 - α21N1) Populations stop growing when: 0 = (K1 - N1 - α12N2) 0 =(K2 - N2 - α21N1) On further rearranging. . . N1 = K1-α12N2 N2 = K2-α21N1 N1 = K1-α12N2 and N2 = K2-α21N1 are really simple equations to describe straight lines! You may recall from high school . . . y=slope(x)+ b N1 is y Effects of species 2 on species 1 (a12) is slope N2 is the x value K1 is y intercept (when x = 0) At every point along these lines growth is stopped Arrow up means growth of species 1 Arrow down means decrease of species 1 The line is called the zero growth isocline for species 1: it represents all combinations of N1 and N2 for which growth of N1 is zero. Any combination of species 1 and 2 below the line means that species 1 will increase Any combination of species 1 and 2 above the line means that species 1 will decrease Similarly. . . . Arrow right means growth of species 2 Arrow left means decrease of species 2 There are four ways to plot these two species together Species 1 increasing while species 2 is decreasing Each red arrow is a vector of the combined direction of both species. Example a:! The species 1 isocline is above the species 2 isocline. Below both isoclines, species 1 and 2 both increase.! In the range of the graph between the two isoclines, we are above the species 2 isocline so it (sp2) decreases, but we are also below the species 1 isocline so it (sp1) continues to increase.! The result is that species 2 declines to zero and species 1 increases to its carrying capacity.! In this case species 1 has competitively excluded species 2. K1>K2/α21 so! K1α21>K2 This means that when species 1 is at its carrying capacity, its impact on species 2 (measured by K1 times a21) is greater than the impact of K2 individuals of species 2.! Thus, species 1 is affecting species 2 more negatively than species 2 affects itself.! Interspecific competition regulates species 2 more than species 2 is regulated by intraspecific competition. Example b:! The species 2 isocline is above the species 1 isocline. From the N2 axis: K2>K1/α12 and so K2α12>K1 K1>K2/α21 so! K1α21>K2 Thus, species 2 is affecting species 1 more negatively than species 1 affects itself.! Example c): Isoclines for the two species cross; the K values on each axis are higher than the K/a values In this situation, from the N1 axis: K1>K2/α21 thus K1α21>K2 Indicating higher impact of interspecific competition than intraspecific competition on species 2 BUT From the N2 axis: K2>K1/α12 so K2α12>K1 Indicating higher impact of interspecific competition than intraspecific competition on species 1. Thus, either species could exclude the other species. Here interspecific competition is stronger than is intraspecific competition. Example d: Isoclines for the two species cross; the K values on each axis are lower than the K/a values From N1 axis: K2/a21>K1 thus K2>K1a21 Indicating that species 2 is regulated more by intraspecific competition than by interspecific competition From the N2 axis: K2<K1/α12 so K2α12<K1 Indicating that species 1 is regulated more by intraspecific competition than by interspecific competition So we can see that when each species is regulated more by intraspecific competition rather than by competition with the other species, the two species can coexist.