"food chains with a scavenger"
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"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
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