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: NEW focus Intraspecific = within species Interspecific = between species Previous focus Scramble & contest density dependence

Competition Types – The Basics Two types of interspecific competition (also saw for intraspecific): Exploitative = indirect competition Competition through a common resource  scramble Interference = direct competition Aggressive / physical encounters for resources  contest Plant competition for nutrients Plant chemical warfare e.g., manzanita

Exploitative Interspecific Competition We will only model exploitative interspecific 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 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 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 http://www.youtube.com/watch?v=EzLADnLsNRs, This doesn’t have to do with victims/prey, but does get to the idea of two predators enter, one predator leaves. 

Exploitative Interspecific Competition 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: dV/dt = bvV - dvV - aVP dP/dt = acVP - dpP 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 Resource, R: dR/dt = brR - drR - aRP Predator, P: dP/dt = acRP - dpP

Competition Equations Resource, R: dR/dt = brR - drR - aRP Predator, P: dP/dt = acRP - dpP Key: Need equations for each consumer! Predator 1, P1: dP1/dt = a1c1RP1 – d1P1 Predator 2, P2: dP2/dt = a2c2RP2 – d2P2 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: Resource, R: dR/dt = brR - drR – a1RP1 – a2RP2 brR Number of resources born per time (same as before) drR Number 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 Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 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 dR/dt = brR - drR – a1RP1 – a2RP2 Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 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: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 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: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 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

Which consumer is the winner? Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) Which consumer is the winner?  Consumer 2

Two consumers introduced Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) Two consumers introduced Consumer 1 dominates at first when resources are abundant Consumer 2 wins in long-run when resources are low Steady state reached

Steady State Winner Resource Consumer 1 Consumer 2 Biomass (g/L) 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 reached

Use our competition equations to derive 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) 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 reached

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* Day 1 . Day 12 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 Rotifer Biomass (μg/L) Algae R* Days

. . Chemostat R* Experiment – Consumer 1 R* Challenge Question: Day 1 . Day 12 Challenge Question: What would happen if we increased the amount of algae being delivered? Think about last class! Rotifer Biomass (μg/L) Algae R* Days

. . Chemostat R* Experiment – Consumer 1 R* Challenge Question: Day 1 . Day 12 Challenge Question: What would happen if we increased the amount of algae being delivered? Algae level stays SAME Rotifer level would increase 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 0 = a1c1R*P1* – d1P1* a1c1R*P1* = d1P1* a1c1R*P1* d1P1* = Predator 1: dP1/dt = a1c1RP1 – d1P1 Let’s look at the other consumer 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* a1c1P1* a1c1R*P1* d1P1* = d1 a1c1 R* =

R* Rule 0 = a1c1R*P1* – d1P1* 0 = a2c2R*P2* – d2P2* a1c1R*P1* = d1P1* Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 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* a1c1P1* a1c1R*P1* d1P1* = a2c2P2* a2c2R*P2* d2P2* = d1 a1c1 R* = d2 a2c2 R* =

R* Rule 0 = a1c1R*P1* – d1P1* a1c1R*P1* = d1P1* a1c1R*P1* d1P1* = Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 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* a1c1P1* a1c1R*P1* d1P1* = Daphnia d1 a1c1 R* = d2 a2c2 R* =

Chemostat R* Experiment – Consumer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Day 21 . Day 1 . Day 12 Daphnia Biomass (μg/L) Daphnia have a R* = 20 μg/L Algae R* Days

R* Rule – Competitive Exclusion Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 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 Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 From the chemostat experiment: Rotifers have a R* = 40 μg/L Daphnia have a R* = 20 μg/L Daphnia wins! Consumer with the lowest R* always wins Rotifers will take early lead, but Daphnia will win at lower resource levels Challenge Question: Given each R*, which consumer wins in long-run? What will happen if these two consumers are put together?

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 dp ac R* = 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

Chemostat R* Experiment – Both Consumers . . . . Day 1 . Day 12 . Day 21 Rotifer Daphnia Biomass (μg/L) Algae RD* Days

. . . . . . Chemostat R* Experiment – Both Consumers RR* RD* Rotifers do best at high resources But when R drops below rotifer R* (due to Daphnia consumption) rotifers decline . Day 1 . Day 12 . Day 21  Daphnia win due to lower R* Rotifer Daphnia Biomass (μg/L) RR* Algae RD* Days

Competitive Exclusion Summary To sum up Given these 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

Adding Type II functional response Stella lab – Meet in Computer Lab Looking Ahead Next Class: More R* Adding Type II functional response Lab Tomorrow Competition modeling Stella lab – Meet in Computer Lab