FW364 Ecological Problem Solving Class 24: Competition November 27, 2013

Recap from Last Class More TODAY dP1/dt = a1c1RP1 – d1P1 Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 From the chemostat experiment: More TODAY 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

. . . . . . 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

Competition Equation Assumptions Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 Additional assumptions (from predator-prey models): 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)

Competition Equation Assumptions Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2 Resource: dR/dt = brR - drR – a1RP1 – a2RP2 Additional assumptions (from predator-prey models): 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)

Adding Consumer Satiation Assumption 4: Consumers are insatiable i.e., consumers eat the same proportion of the resource population (a) no matter how many resources (R) there are  Type I functional response To relax assumption, we can make the consumer feeding rate (aR) a saturating function of the resource abundance  Type II functional response R aR low many high Type I functional response (linear) Type II functional response Satiation Let’s define an equation for Type II response

Adding Consumer Satiation First, we need a new symbol for feeding rate: Feeding rate: f For a Type I functional response (linear): f = aR For a Type II functional response (saturating): fmax R R + h f = Where: fmax is the maximum feeding rate h is the half-saturation constant R is resource abundance Let’s look at a figure…

Adding Consumer Satiation fmax R R + h f = Where: fmax is the maximum feeding rate h is the half-saturation constant R is resource abundance fmax = 5 Consumer feeding rate approaches fmax at high resource abundance

Adding Consumer Satiation What is h for this figure? fmax R R + h f = Where: fmax is the maximum feeding rate h is the half-saturation constant R is resource abundance fmax = 5 Challenge Question: What is h for this figure? h is the value of R when the feeding rate is half of the maximum value i.e., h is value of R when f/fmax = 0.5

Adding Consumer Satiation So, h is value of R when f is 2.5 fmax R R + h f = Where: fmax is the maximum feeding rate h is the half-saturation constant R is resource abundance fmax = 5 fmax = 5 and half of 5 is 2.5 So, h is value of R when f is 2.5  h = 2 2 h is the value of R when the feeding rate is half of the maximum value i.e., h is value of R when f/fmax = 0.5

Adding Consumer Satiation fmax R R + h f = Where: fmax is the maximum feeding rate h is the half-saturation constant R is resource abundance fmax = 5 A Type II functional response can apply to any type of consumer: Carnivores, herbivores, parasites, and plants Though plants do not eat (attack) resources, their growth still increases with resource abundance to some threshold rate (i.e., until saturated with resources) Let’s put the Type II response into our consumer growth equation (dP/dt) h is the value of R when the feeding rate is half of the maximum value i.e., h is value of R when f/fmax = 0.5

Type II Functional Response - Equation Type I functional response: dP/dt = acRP – dpP Re-arrange to get aR adjacent: dP/dt = caRP – dpP Replace aR with f: dP/dt = cfP – dpP General equation that we can put any functional response (f) into: dP/dt = cfP – dpP fmax R R + h f = With Type II functional response: cfmax RP R + h dP/dt = – dpP Plug f into general equation:  Equation for consumer growth with a Type II functional response

R* for Type II Functional Response Our functional response has changed, so we need to a new R* equation i.e., R* for Type II response cfmax RP R + h dP/dt = – dpP R* occurs at steady-state, so set dP/dt = 0 cfmax R*P* R* + h 0 = – dpP* Solve for R* cfmax R*P* R* + h = dpP* dp h c fmax - dp R* = …a whole lot of algebra you do in Lab 10…

R* for Type II Functional Response dp h c fmax - dp R* = Conclusions: With a Type II functional response: R* depends on consumer death rate, half saturation constant, conversion efficiency, and max feeding rate If consumer death rate increases, R* increases If consumer half saturation constant increases, R* increases If conversion efficiency increases, R* decreases If max feeding rate increases, R* decreases

Saturation & Consumer Birth Rate That was a lot about feeding rate… … need to get back to competition To do that, need to make a crucial link between consumer feeding rate and birth rate R* is key for competition… and R* depends on dp Competition winner is the consumer alive at steady state … i.e., when bp = dp dp h c fmax - dp R* =  Knowing birth rate of consumer is important for determining competition outcome Let’s look at how a saturating feeding rate affects consumer birth rate

Saturation & Consumer Birth Rate cfmax RP R + h dP/dt = – dpP Type II functional response: cfmax R R + h dP/dt = P – dpP Minor re-arrangement: This is all equivalent to our consumer birth rate i.e., consumers are born by feeding on prey Consumer birth rate function should curve the same as the feeding rate, since birth rate is just feeding rate multiplied by a constant (conversion efficiency)

Saturation & Consumer Birth Rate Resource abundance (R) high fmax Feeding rate (f) Resource abundance (R) high bmax high Resource abundance (R) birth rate (bp)

Saturation & Consumer Birth Rate high fmax Consumer birth rate increases with resource abundance to a threshold rate, bmax (threshold birth rate is due to feeding rate hitting threshold) h, the half-saturation constant, still applies: h is the value of R when the birth rate is half of the maximum value Feeding rate (f) Resource abundance (R) high high bmax birth rate (bp) Resource abundance (R) high

Saturation & Consumer Death Rate So that’s how consumer birth rate changes with resource density… …now on to death rate We have been making an (implicit) assumption about how consumer death rate changes with resource density cfmax RP R + h dP/dt = – dpP We’ve been assuming that the consumer death rate is a constant (dp) i.e., that the consumer death rate does NOT change with resource density To plot this assumption on a figure…

Saturation & Consumer Death Rate Consumer death rate is just a straight line at any value along the y-axis high Death rate death rate (dp) Resource abundance (R) high If we combine the death rate function with the birth rate curve…

Saturation & Consumer Death Rate Consumer death rate is just a straight line at any value along the y-axis high Birth rate Death rate birth rate (bp) death rate (dp) Resource abundance (R) high If we combine the death rate function with the birth rate curve… we have a useful trick for graphically determining R* for a consumer… (consumer birth rate and death rate must be plotted on the same scale!)

Graphical approach to R* high Birth rate Death rate birth rate (bp) death rate (dp) Resource abundance (R) high Challenge question: A special point on this figure represents steady state… Where is this point?

Graphical approach to R* high Birth rate Death rate birth rate (bp) death rate (dp) Steady state when b = d Resource abundance (R) high Challenge question: A special point on this figure represents steady state… Where is this point?

Graphical approach to R* high Birth rate Death rate birth rate (bp) death rate (dp) Steady state when b = d R* Resource abundance (R) high KEY feature of this graph: The resource abundance (i.e., value on x-axis) at the steady state point (i.e., intersection of b and d functions) is R*!

Graphical approach to R* Resource abundance (R) high Birth rate Death rate birth rate (bp) death rate (dp) Steady state when b = d R* Resource abundance (R) high Key application: If we plot the birth and death rates of two competing species on same figure, we can determine which consumer will win based on who has the lower R*

Graphical Approach to R* First, one more question for single consumer: Birth rate high Resource abundance (R) birth rate (bp) Death rate death rate (dp) Quick Challenge Question: What happens if the death rate is higher than the birth rate?

Graphical Approach to R* First, one more question for single consumer: Birth rate high Resource abundance (R) birth rate (bp) Death rate death rate (dp) What happens if the death rate is higher than the birth rate?  Consumer goes extinct, even without the competitor Now let’s look at resource competition

Graphical R* & Competition Outline: Look at four graphical cases of two-species competition (competition with Type II functional response) Consumers will have: Case 1: Different birth rates, same death rate and h Case 2: Different birth rates and death rates, same h Case 3: Different birth rates, same death rates, different h Case 4: Different birth rates, same death rate, different h w/ twist For each case, we’ll determine competition winner

Case 1: Effect of different birth rates d1d2 Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Both consumers have same death rate, d1 = d2 Monod curves never cross Who wins?

Case 1: Effect of different birth rates d1d2 R1* R2* Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Both consumers have same death rate, d1 = d2 Monod curves never cross Consumer 1 wins: R1* < R2*  Higher birth rate makes better competitor

Case 2A: Effect of different death rates b1 d1 b2 d2 Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Consumer 1 has a higher death rate than Consumer 2 (d1 > d2) Monod curves never cross Who wins?

Case 2A: Effect of different death rates b1 d1 R1* b2 d2 R2* Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Consumer 1 has a higher death rate than Consumer 2 (d1 > d2) Monod curves never cross Consumer 2 wins: R2* < R1*  Lower death rate makes better competitor

Case 2B: Effect of different death rates Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Consumer 2 has a higher death rate than Consumer 1 (d2 > d1) Monod curves never cross Who wins?

Case 2B: Effect of different death rates Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2) Consumer 2 has a higher death rate than Consumer 1 (d2 > d1) Monod curves never cross Consumer 1 wins: R1* < R2*  Lower death rate makes better competitor

Case 3: Effect of different h At very high resource density b1b2 b1 At very high resource density b2 d1d2 Both consumers have same maximum birth rate, b1max = b2max Both consumers have same death rate, d1 = d2 Consumer 1 has a lower h ( Consumer 1 approaches bmax at lower R) Who wins?

Case 3: Effect of different h b1b2 b1 At very high resource density b2 d1d2 R1* R2* Both consumers have same maximum birth rate, b1 = b2 Both consumers have same death rate, d1 = d2 Consumer 1 has a lower h ( Consumer 1 approaches bmax at lower R) Consumer 1 wins: R1* < R2*  Lower h makes better competitor

Graphical R* & Competition Summary What makes a better competitor (i.e., lower R*)? Higher birth rate Lower death rate Lower h dp h c fmax - dp R* = We reached the same conclusions looking at R* equation If consumer death rate increases, R* increases  so lower dp makes better competitor If consumer half saturation constant increases, R* increases  so lower h makes better competitor If conversion efficiency increases, R* decreases If max feeding rate increases, R* decreases  Birth rate is just conversion efficiency * feeding rate, so higher birth rate makes better competitor

Graphical R* & Competition Summary What makes a better competitor (i.e., lower R*)? Higher birth rate Lower death rate Lower h Does this perfect competitor exist in nature? Not really… there are always trade-offs in nature e.g., high max birth rate requires more resources, foraging exposes consumers to predation, and so high bmax associated with high death rates High birth rate rabbit takes risks to forage