"Food Chains with a Scavenger"Penn State Behrend

Summer 2005 REU --- NSF/ DMS #0236637 Ben Nolting, Joe Paullet, Joe Previte

Tlingit Raven, an important scavenger in arctic ecosystems

R.E.U.?

• Research Experience for Undergraduates• Usually a summer • 100’s of them in science (ours is in math

biology)• All expenses paid plus stipend $$$!• Competitive• Good for resume • Experience doing research

Scavengers: Animals that subsist primarily on carrion (the bodies of deceased animals)

Ravens

Beetles

Crabs

Hyenas, Wolves, and Foxes

Vultures

Earwigs

• e.g., x= hare;• y =lynx (fox)

Introduce scavenger on a simple Lotka-Volterra Food Chain

Lotka – Volterra 2- species model

(1920’s A.Lotka & V.Volterra)• dx/dt = ax-bxy

dy/dt = -cx+dxy

a → growth rate for xc → death rate for yb → inhibition of x in presence of yd → benefit to y in presence of x

• Want DE to model situation

Analysis of 2-species model

• Solutions follow

a ln y – b y + c lnx – dx=C

Nk

xbaxxN

i

n

jjijiik

,,1

,)(1 1

More general systems of this type look like:

1. Quadratic (only get terms like xixj)

2. Studied to death! But still some open problems (another talk)

Volterra Proved:

T

TTxdttx

0*

1 )(lim

If there is an interior fixed point with x-coord x* :

Similar with others coordinates (we’ll use this later)

Simple Scavenger Model

lynx

hare

beetle

Among other things, a scavenger species z should benefit whenever a predator kills its prey (scavenger eats dead body)

xyz is proportional to the number of interactions between scavengers and carrion.

hyzgxzfxyzezzdxycyybxyaxx

The Simple Scavenger Model

Note: To simplify the analysis of these systems, it is often convenient to rescale parameters.

The number of parameters that you can eliminate depends on the structure of the system.

hyzgxzfxyzezzdxycyybxyaxx

byYdxXat ,,

1,1,1 dba

Results for the simple scavenger system

Three cases:

hgccfehgccfehgccfe ,,

hyzgxzfxyzezzxycyy

xyxx

Fixed point in 2d system: (c,1)

Dynamics trapped on cylinders

Scavenger dies e>cf+gc+h

Scavenger stays boundede = cf+gc+h

Scavenger blows upe<gc+fc+h

Case 1:

z2 = z1

Main Idea: (return map in z) of PROOF

Case 2:

z2> z1 => z3>z2

z3<z2 no good!

z_i monotone increasing

So…

• z1 < z2 => zi increasing

• z1 > z2 => zi decreasing

• z1 = z2 => zi constant (periodic)

Monotone Sequence Theorem: zi either converges or

goes to +∞

Let (x0,y0,z0) be given having period T in the plane

T

TT

T

cdtxy

or

dtxycdtxyx

so

xTxdtx

0

00

0

' 0)0()( Why?

)0()()0()(

)0()(

zTzincreaseszhgcfceifzTzdecreaseszhgcfceif

zTzperiodichgcfceif

i

i

hgcfce

dthygxfxyeT

dtzz

T

TT

00

' 11

))0(/)(ln(1 zTzT

also

Biologists Not Not Pleased!!

I’m NOT pleased

Scavenger dies or blows up except on a set of measure zero!

We want stable behavior,So let’s make the growth of x logisticlogistic:

hyzgxzfxyzezzxycyybxxyxx

2

Know (x,y) -> (c, 1-bc) use this to see

e<f(1-bc)c+gc+h(1-bc) implies z is unbounded

e>f(1-bc)c+gc+h(1-bc) implies z goes extinct

e=f(1-bc)c+gc+h(1-bc) implies z to a non-zero limit

Still No good!

Let’s go back to LV w/o logistic,

But put a quadratic death term on the scavenger.

2jzhyzgxzfxyzezz

xycyyxyxx

Rutter’s slide

zzz

Average death rate proportional to z, so

Adding a quadratic death term makes perfect sense and is not overkill (but needed here!)

Globally stable limit cycles on every cylinder!

No blow ups or extinctions.

Keys to proof:1) Orbits are confined to cylinders2) For a particular cylinder, the z nullcline

intersects the cylinder at a high point z*.

3) z* is an upper bound for trajectories starting below z*.

4) Every trajectory starting above z* must eventually venture below z*.

5) Very close to xy plane, return map is increasing.

6) Monotone sequence bounded above-> limit.

7) Time averages show you can’t have two limit cycles on the same cylinder.

Other possibilities for further research

• 3 species models w/ scavenger• Scavengers affect other species (crowding)• Scavenger Ring models• More quadratic death terms• Etc. etc. etc.• Ben Nolting (Alaska)

Ring Model