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FW364 Ecological Problem Solving. Class 23: Competition. November 25, 2013. Outline for Today. Shifting focus from predator-prey interactions to two species competition Objectives for Today : Derive equations for two-species resource competition
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FW364 Ecological Problem Solving Class 23: Competition November 25, 2013
Outline for Today • Shifting focus from predator-prey interactions • to two species competition • Objectives for Today: • Derive equations for two-species resource competition • Introduce R* rule to determine competition winner • Objectives for Next Three Classes: • Explore R* rule in more detail • Include Type II functional response for consumers • Examine graphical approaches for determining competition winner • Discuss limits to competitive exclusion • Discuss practical application of resource competition models • No textbook chapters!
Competition Types – The Basics Two types of competition: Intraspecific = within species Interspecific = between species Previous focus Scramble & contest density dependence NEW focus
Competition Types – The Basics Two types of interspecific competition (also saw for intraspecific): Interference = direct competition Aggressive / physical encounters for resources contest Exploitative= indirect competition Competition through a common resource scramble Plant competition for nutrients Plant chemical warfare e.g., manzanita
Exploitative Interspecific Competition We will only model exploitativeinterspecific competition a.k.a. resource competition Conceptual framework Consumer 1 Consumer 2 Resource Applies to: Examples: Carnivores-animal prey Herons and cranes competing for fish in swamps Herbivores-plant Zebras and wildebeest consuming grasses Parasites-host Sea lamprey and copepods parasitizing lake trout Plants-resource Ferns and grass competing for nutrients
Exploitative Interspecific Competition C1 C2 • Key features: R • Competition is an extension of predator-prey concepts we just studied! • Two coupled predator-prey interactions that share same prey (resource) • Interaction between consumers mediated through resource • No direct interaction (this is an assumption!) • Only one consumer can persist at steady state (another assumption) • Competitive exclusion principle • i.e., when two species compete over a common resource, only one species (the superior competitor) can persist in the long-term
Exploitative Interspecific Competition C1 C2 • Key features: R • To reiterate: • With the competitive exclusion principle, • we are assuming there is always a winner in the long run • i.e., one consumer will out-compete (exclude) the other • The winner of the competition is the consumer that is still alive at steady state, whereas the loser is the consumer that has gone extinct
Exploitative Interspecific Competition C1 C2 • Goals R • Determine the competition winner at steady state (algebra) • Describe dynamics and forecast time to extinction of inferior competitor (Stella – Lab 10) • Important to understand the difference between the two approaches • and how they complement each other • #1 tells us WHO will win • #2 tells us WHEN the loser goes extinct, • and HOW populations change through time • Let’s build equations!
Competition Equations • Like before, we’ll build simple models that capture the essence • of two-species resource competition • (we’ll be making many assumptions) • Start with our basic (coupled) predator-prey equations: • dP/dt= acVP - dpP • dV/dt= bvV- dvV - aVP • Victim, V: • Predator, P: • Adapt more general notation of consumer-resource • substitute R for V, but keep P • Note that I will use “consumer” and “predator” interchangeably • dP/dt= acRP - dpP • Resource, R: • Predator, P: • dR/dt= brR- drR - aRP
Competition Equations • dR/dt= brR- drR - aRP • dP/dt= acRP - dpP • Resource, R: • Predator, P: • Key: Need equations for each consumer! • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1, P1: • Predator 2, P2: • Each consumer can have own a, c, and d (note subscripts specific to consumers) • This will be important later for determining competition winner • Now need to include consumption by both consumers in resource equation: • dR/dt= brR- drR – a1RP1 – a2RP2 • Resource, R: • brR Number of resources born per time (same as before) • drRNumber of non-predatory resource deaths per time (same as before) • a1RP1 Number of resources eaten by Consumer 1 population per time • a2RP2 Number of resources eaten by Consumer 2 population per time
Competition Equations • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • dR/dt= brR- drR – a1RP1 – a2RP2 • Resource: • These are our competition equations! • Some key features: • Equations are adaptable to competition between plants for resources • with slight modification to the resource equation • We can predict the outcome of competition from the consumer equations! • (do not need the resource equation)
Competition Equations • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • dR/dt= brR- drR – a1RP1 – a2RP2 • Resource: • Assumptions: • The consumer populations cannot exist if there are no resources • In the absence of both consumers, the resources grow exponentially • Consumers encounter prey randomly (“well-mixed” environment) • Consumers are insatiable (Type I functional response) • No age / stage structure • Consumers do not interact with each other except through consumption • (i.e., exploitative competition)
Steady State Winner • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • dR/dt= brR- drR – a1RP1 – a2RP2 • Resource: • Most important question for competition: • Who will win in the long-term? • i.e., which competitor will be alive once the system has reached steady state? • Key Point: • The competitor that wins in the long-term MAY NOT be the • competitor that does best initially • Different factors are involved in short-term vs. long term competitive ability
Steady State Winner • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • dR/dt= brR- drR – a1RP1 – a2RP2 • Resource: • Short vs. long-term competitive ability • We all know an example of this: • One consumer may have advantage • of speed in the short-term, • but endurance is what matters in • the long-run for competition • Let’s look at a figure The tortoise and the hare fable
Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) • Consumer 2 • Which consumer is the winner?
Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) • Two consumers introduced • Consumer 2 wins in long-run when resources are low • Steady state reached • Consumer 1 dominates at first when resources are abundant
Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) • Steady state reached • Reminder: Steady state is when there is no change • in abundance through time • i.e., dP1/dt = 0, dP2/dt= 0, and dR/dt = 0
Steady State Winner Resource Consumer 1 Consumer 2 • Next step: • Use our competition equations to derive • general equations that tell us which consumer will survive and which consumer will go extinct • at steady state • R* rule Biomass (g/L) • Steady state reached • Reminder: Steady state is when there is no change • in abundance through time • i.e., dP1/dt = 0, dP2/dt= 0, and dR/dt = 0
R* Rule • R* Rule determines competitive dominance at steady state • R* is the resource level at equilibrium • The R* Rule: • The resource abundance at equilibrium (R*) is that level of the resource at which the consumer is just maintained in the system • i.e., R* is the lowest resource level at which a consumer can be sustained • If the resource level were to decrease, the consumer would go extinct • Each consumer has it’s own R* • Competing consumers will (almost always) have different R* • R* will determine which consumer wins
R* Rule • R* Rule determines competitive dominance at steady state • R* is the resource level at equilibrium • The R* Rule: • The resource abundance at equilibrium (R*) is that level of the resource at which the consumer is just maintained in the system • i.e., R* is the lowest resource level at which a consumer can be sustained • If the resource level were to decrease, the consumer would go extinct • R* for a • rotifer consumer • Let’s look at an experimental example
Chemostat R* Experiment – Consumer 1 . . . . . . . . • R* is the lowest level of algae that maintains rotifers in the system • = steady state abundance (biomass) of algae • For rotifers, R* = 40 μg/L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 1 Day 12 . . Rotifer Biomass (μg/L) Algae R* Days
Chemostat R* Experiment – Consumer 1 . . . . . . . . . . . . • Challenge Question: • What would happen if we increased the amount of algae being delivered? • Think about last class! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 1 Day 12 . . Rotifer Biomass (μg/L) Algae R* Days
Chemostat R* Experiment – Consumer 1 . . . . . . . . . . . . • Challenge Question: • What would happen if we increased the amount of algae being delivered? • Algae level stays SAME • Rotifer level would increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 1 Day 12 . . Rotifer Biomass (μg/L) Algae R* Days
R* Rule • That’s how we can determine R* empirically (experimentally) for a single species • Can also determine R* by building an equation • Let’s derive equation for R* • Recall for predator-prey equations that we used the • predator equation to derive equation for V* • and prey equation to derive equation for P* • Likewise, we will use consumer equation to derive equation for R* • We’ll start with Consumer 1
R* Rule • dP1/dt= a1c1RP1 – d1P1 • Let’s look at the other consumer • Predator 1: • R* occurs at steady-state, • so set dP1/dt = 0 • and denote equilibrium with * • Conclusions: • The minimum resource requirement • (R*) for a consumer is determined by • the consumer death rate, attack rate, • and conversion efficiency • If death rate increases, R* increases • If attack rate increases, R* decreases • If conversion efficiency increases, • R* decreases • Note: • We determine R* for each consumer • when ALONE! • 0 = a1c1R*P1* – d1P1* • Solve for R* • a1c1R*P1*= d1P1* • d1P1* • a1c1R*P1* • = • a1c1P1* • a1c1P1* • d1 • R*= • a1c1
R* Rule • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • R* occurs at steady-state, • so set dP1/dt = 0 • and denote equilibrium with * • R* occurs at steady-state, • so set dP2/dt = 0 • and denote equilibrium with * • 0 = a1c1R*P1* – d1P1* • 0 = a2c2R*P2* – d2P2* • Solve for R* • Solve for R* • a1c1R*P1*= d1P1* • a2c2R*P2*= d2P2* • d2P2* • d1P1* • a2c2R*P2* • a1c1R*P1* • = • = • a1c1P1* • a2c2P2* • a2c2P2* • a1c1P1* • d2 • d1 • R*= • R*= • a1c1 • a2c2
R* Rule • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • R* occurs at steady-state, • so set dP1/dt = 0 • and denote equilibrium with * • Let’s look at a • chemostat experiment for • the second consumer • 0 = a1c1R*P1* – d1P1* • Solve for R* • a1c1R*P1*= d1P1* • d1P1* • a1c1R*P1* • Daphnia • = • a1c1P1* • a1c1P1* • d1 • d2 • R*= • R*= • a1c1 • a2c2
Chemostat R* Experiment – Consumer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 1 . Day 12 Day 21 . . . Daphnia Biomass (μg/L) • Daphnia have a R* = 20 μg/L Algae R* Days
R* Rule – Competitive Exclusion • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • From the chemostat experiment: • Rotifers have a R* = 40 μg/L • Daphnia have a R* = 20 μg/L • Challenge Question: • Given each R*, which consumer wins in long-run? • What will happen if these two consumers are put together?
R* Rule – Competitive Exclusion • dP1/dt= a1c1RP1 – d1P1 • dP2/dt= a2c2RP2 – d2P2 • Predator 1: • Predator 2: • From the chemostat experiment: • Rotifers have a R* = 40 μg/L • Daphnia have a R* = 20 μg/L • Challenge Question: • Given each R*, which consumer wins in long-run? • What will happen if these two consumers are put together? • Daphnia wins! • Consumer with the lowest R* always wins • Rotifers will take early lead, but Daphnia will win at lower resource levels
More R* • When the resource level falls below the equilibrium level for a consumer • (when R < R*), • the consumer density will decline • When each consumer is alone, the consumer will drive R down to R*, • but when a competitor is added, the second consumer can drive R < R*! • The consumer whose biological characteristics are such that its minimum resource requirement (R*) are lowest WINS competition • According to our R* equation: • The consumer with a lower death rate, higher attack rate, • and/or greater conversion efficiency will win • i.e., any characteristic that decreases R* will provide a competitive advantage • dp • R*= • ac
Chemostat R* Experiment – Both Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 1 . Day 12 Day 21 . . . Rotifer Daphnia Biomass (μg/L) Algae RD* Days
Chemostat R* Experiment – Both Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Rotifers do best at high resources . . . . . . . . . . . . . . . . . . • But when R drops below rotifer R* • (due to Daphnia consumption) • rotifers decline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Daphnia win due to lower R* . . . . . . . Day 1 . Day 12 Day 21 . . . Rotifer Daphnia Biomass (μg/L) RR* Algae RD* Days
Competitive Exclusion Summary • To sum up • Giventhese assumptions: • a stable environment • competitors that are not equivalent (different R*) • a single resource • unlimited time • Then: • The species with the lowest minimum resource requirement (R*) • will eventually exclude all other competitors • Let’s look at some of the other assumptions we have made more closely
Looking Ahead Next Class: More R* Adding Type II functional response Lab Tomorrow Competition modeling Stella lab – Meet in Computer Lab