FW364 Ecological Problem Solving

Class 24: CompetitionNovember 27, 2013

Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2

From the chemostat experiment:

Daphnia wins!Consumer with the lowest R* always wins

Rotifers will take early lead, but Daphnia will win at lower resource levels

Daphnia have a R* = 20 μg/LRotifers have a R* = 40 μg/L

More TODAY

Recap from Last Class

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Daphnia

Rotifer

Rotifers do best at high resources

Daphnia win due to lower R*

But when R drops below rotifer R*(due to Daphnia consumption)

rotifers decline

Chemostat R* Experiment – Both Consumers

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

Additional assumptions (from predator-prey models):

1. The consumer populations cannot exist if there are no resources2. In the absence of both consumers, the resources grow exponentially3. Consumers encounter prey randomly (“well-mixed” environment)4. Consumers are insatiable (Type I functional response)5. No age / stage structure6. Consumers do not interact with each other except through consumption (i.e., exploitative competition)

Resource: dR/dt = brR - drR – a1RP1 – a2RP2

Competition Equation Assumptions

Additional assumptions (from predator-prey models):

1. The consumer populations cannot exist if there are no resources2. In the absence of both consumers, the resources grow exponentially3. Consumers encounter prey randomly (“well-mixed” environment)4. Consumers are insatiable (Type I functional response)5. No age / stage structure6. Consumers do not interact with each other except through consumption (i.e., exploitative competition)

Resource: dR/dt = brR - drR – a1RP1 – a2RP2

Competition Equation Assumptions

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

0 many

high Type I functional response (linear)

Type II functional response

Satiation

To relax assumption, we can make the consumer feeding rate (aR)a saturating function of the resource abundance Type II functional response

Adding Consumer Satiation

Let’s define an equation for Type II response

Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance

fmax RR + h

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):

Let’s look at a figure…

fmax RR + h

0 5 10 15 20 25 300

Resource abundance (R)

(f) fmax = 5

Consumer feeding rate approaches fmax at high

resource abundance

fmax RR + h

0 5 10 15 20 25 300

(f) fmax = 5

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

Challenge Question:

What is h for this figure?

fmax RR + h

0 5 10 15 20 25 300

(f) fmax = 5

fmax = 5 and half of 5 is 2.5

So, h is value of R when f is 2.5 h = 2

fmax RR + h

0 5 10 15 20 25 300

(f) 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)

Type II Functional Response - Equation

dP/dt = acRP – dpPType I functional response:

fmax RR + h

dP/dt = caRP – dpPRe-arrange to get aR adjacent:

Replace aR with f: dP/dt = cfP – dpP

With 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 + h

dP/dt = – dpP

Equation for consumer growth with a Type II functional response

cfmax RPR + h

dP/dt = – dpPOur functional response has changed,

so we need to a new R* equationi.e., R* for Type II response

cfmax R*P*R* + h

0 = – dpP*

R* occurs at steady-state,so set dP/dt = 0

cfmax R*P*R* + h

= dpP*

…a whole lot of algebra you do in Lab 10…

dp hc fmax - dp

Solve for R*

R* for Type II Functional Response

dp hc fmax - dp

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* decreases

R* for Type II Functional Response

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 linkbetween consumer feeding rate and birth rate

R* 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 outcome

Let’s look at how a saturating feeding rate affects consumer birth rate

dp hc fmax - dp

cfmax RPR + h

dP/dt = – dpPType II functional response:

Minor re-arrangement:cfmax RR + h

dP/dt = P – dpP

This is all equivalent to our consumer birth ratei.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)

0 5 10 15 20 25 30012345 fmax

0 5 10 15 20 25 30012345 bmax

0 5 10 15 20 25 30012345 fmax

0 5 10 15 20 25 30012345 bmax

p)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

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 abouthow consumer death rate changes with resource density

cfmax RPR + 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…

0 5 10 15 20 25 30012345

Consumer death rate is just a straight line at any value along the y-axis

If we combine the death rate function with the birth rate curve…

Death rate

Consumer death rate is just a straight line at any value along the y-axis

If we combine the death rate function with the birth rate curve…we have a useful trick for graphically determining R* for a consumer…