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FW364 Ecological Problem Solving . Class 24: Competition. November 27, 2013. Recap from Last Class. dP 1 / dt = a 1 c 1 RP 1 d 1 P 1. dP 2 / dt = a 2 c 2 RP 2 d 2 P 2. Predator 1:. Predator 2:. From the chemostat experiment:. More TODAY. Rotifers have a R* = 40 g/L. - PowerPoint PPT Presentation

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FW364 Ecological Problem Solving Class 24: CompetitionNovember 27, 20131Predator 1:dP1/dt = a1c1RP1 d1P1Predator 2:dP2/dt = a2c2RP2 d2P2

From the chemostat experiment:

Daphnia wins!

Consumer with the lowest R* always winsRotifers will take early lead, but Daphnia will win at lower resource levelsDaphnia have a R* = 20 g/LRotifers have a R* = 40 g/LMore TODAYRecap from Last Class2RD*................................Biomass (g/L)....................................Day 1

.........Day 12

...........Day 21

DaphniaAlgaeDaysRotiferRotifers do best at high resources Daphnia win due to lower R*But when R drops below rotifer R*(due to Daphnia consumption)rotifers declineRR*Chemostat R* Experiment Both Consumers3Competitive Exclusion SummaryTo sum up

Given these assumptions:a stable environmentcompetitors that are not equivalent (different R*)a single resourceunlimited time

Then:The species with the lowest minimum resource requirement (R*) will eventually exclude all other competitorsLets look at some of the other assumptions we have made more closelyAdditional 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)Predator 1:dP1/dt = a1c1RP1 d1P1Predator 2:dP2/dt = a2c2RP2 d2P2Resource:dR/dt = brR - drR a1RP1 a2RP2Competition Equation Assumptions5Additional 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)Predator 1:dP1/dt = a1c1RP1 d1P1Predator 2:dP2/dt = a2c2RP2 d2P2Resource:dR/dt = brR - drR a1RP1 a2RP2Competition Equation Assumptions6Assumption 4: Consumers are insatiablei.e., consumers eat the same proportion of the resource population (a) no matter how many resources (R) there are Type I functional responseRaRlow0manyhighType I functional response (linear)Type II functional responseSatiationTo relax assumption, we can make the consumer feeding rate (aR)a saturating function of the resource abundance Type II functional responseAdding Consumer SatiationLets define an equation for Type II response7Adding Consumer SatiationWhere:fmax is the maximum feeding rateh is the half-saturation constantR is resource abundancefmax RR + hf =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):Lets look at a figure8Adding Consumer SatiationWhere:fmax is the maximum feeding rateh is the half-saturation constantR is resource abundancefmax RR + hf =fmax = 5Consumer feeding rate approaches fmax at high resource abundance9Adding Consumer SatiationWhere:fmax is the maximum feeding rateh is the half-saturation constantR is resource abundancefmax RR + hf =fmax = 5h 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 = 0.5Challenge Question:

What is h for this figure?10Adding Consumer SatiationWhere:fmax is the maximum feeding rateh is the half-saturation constantR is resource abundancefmax RR + hf =fmax = 5h 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 = 0.5fmax = 5 and half of 5 is 2.5

So, h is value of R when f is 2.5

h = 2211Adding Consumer SatiationWhere:fmax is the maximum feeding rateh is the half-saturation constantR is resource abundancefmax RR + hf =fmax = 5h 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 = 0.5A 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)

Lets put the Type II response into our consumer growth equation (dP/dt)12Type II Functional Response - EquationdP/dt = acRP dpPType I functional response:fmax RR + hf =dP/dt = caRP dpPRe-arrange to get aR adjacent:Replace aR with f:dP/dt = cfP dpPWith Type II functional response: Plug f into general equation:dP/dt = cfP dpPGeneral equation that we can put any functional response (f) into:cfmax RPR + hdP/dt = dpP Equation for consumer growth with a Type II functional response13cfmax RPR + hdP/dt = dpPOur functional response has changed,so we need to a new R* equationi.e., R* for Type II responsecfmax R*P*R* + h0 = dpP*R* occurs at steady-state,so set dP/dt = 0cfmax R*P*R* + h= dpP*a whole lot of algebra you do in Lab 10dp hc fmax - dpR* =Solve for R*R* for Type II Functional Response14dp hc fmax - dpR* =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* increasesIf consumer half saturation constant increases, R* increasesIf conversion efficiency increases, R* decreasesIf max feeding rate increases, R* decreasesR* for Type II Functional Response15Saturation & Consumer Birth RateThat was a lot about feeding rate need to get back to competition

To do that, need to make a crucial linkbetween consumer feeding rate and birth rateR* is key for competition and R* depends on dp

Competition winner is the consumer aliveat steady state i.e., when bp = dp Knowing birth rate of consumer is important for determining competition outcomeLets look at how a saturating feeding rate affects consumer birth ratedp hc fmax - dpR* =16Saturation & Consumer Birth Ratecfmax RPR + hdP/dt = dpPType II functional response:Minor re-arrangement:cfmax RR + hdP/dt =P dpPThis is all equivalent to our consumer birth ratei.e., consumers are born by feeding on preyConsumer birth rate function should curve the same as the feeding rate,since birth rate is just feeding rate multiplied by a constant(conversion efficiency) 17Saturation & Consumer Birth RatefmaxhighhighResource abundance (R)Feeding rate (f)bmaxhighhighResource abundance (R)birth rate (bp)18Saturation & Consumer Birth RatefmaxhighhighResource abundance (R)Feeding rate (f)bmaxhighhighResource abundance (R)birth rate (bp)Consumer birth rate increases with resource abundanceto 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 value19Saturation & Consumer Death RateSo thats how consumer birth rate changes with resource densitynow on to death rateWe have been making an (implicit) assumption abouthow consumer death rate changes with resource densitycfmax RPR + hdP/dt = dpPWeve 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 figure20highhighResource abundance (R)Saturation & Consumer Death Ratedeath rate (dp)Consumer death rate is just a straight line at any value along the y-axisIf we combine the death rate function with the birth rate curveDeath rate21Saturation & Consumer Death RateConsumer death rate is just a straight line at any value along the y-axisIf we combine the death rate function with the birth rate curvewe have a useful trick for graphically determining R* for a consumerBirth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)(consumer birth rate and death rate must be plotted on the same scale!)22Birth ratehighhighResource abundance (R)birth rate (bp)Graphical approach to R*Death ratedeath rate (dp)Challenge question:

A special point on this figure represents steady stateWhere is this point?23Birth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)Challenge question:

A special point on this figure represents steady stateWhere is this point?Steady state when b = dGraphical approach to R*24Birth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)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*!R*Steady state when b = dGraphical approach to R*25Birth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)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* R*Steady state when b = dGraphical approach to R*26Graphical Approach to R*First, one more question for single consumer:Birth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)Quick Challenge Question:

What happens if the death rate is higher than the birth rate?27Graphical Approach to R*First, one more question for single consumer:Birth ratehighhighResource abundance (R)birth rate (bp)Death ratedeath rate (dp)What happens if the death rate is higher than the birth rate?

Consumer goes extinct, even without the competitorNow lets look at resource competition28Graphical R* & Compe