MATHEMATICAL MODEL WITH
DIATOMS, COROPHIUM AND
SNAILS
Department of Mathematics and Statistics
University of New Brunswick
Hu Xi, James Watmough and Wang Lin
OUTLINE
1.
2.
3.
4.
5.
6.
Introduction
Objectives
Model
Dynamics
Interpretation
Next Steps
INTRODUCTION
1.
2.
3.
Both the amphipods Corophium volutator and
mud snails Ilynassa obsoleta feed on diatoms
Snails also feed on Corophium
Diatoms, Corophium and snails form a
competitive, predator-prey relationship
(i.e. intraguild predation)
OBJECTIVES
1.
2.
3.
Formulate a mathematical model based on
the relationship among diatoms,Corophium
and snails.
Analyze the model and find out the conditions
under which both diatoms and Corophium will
persist in the system.
Assess how an increasing population of snails
may affect populations of Corophium and
diatoms.
DIATOM DYNAMICS
Logistic growth
Functional responses of
Corophium and snails
dN
N
A1CN
A2 SN
 rN (1 
)

dt
K
N  B1
N  B2
N: abundance of diatoms
C: abundance of Corophium
S: abundance of snails
DIATOM-COROPHIUM DYNAMICS
dN
N
A1CN
A2 SN
 rN (1 
)

dt
K
N  B1
N  B2
dC
eA1CN

 (d  d 2 S )C
dt
N  B1
Birth
Death
N: abundance of diatoms (chlorophyll a (mg/m2) )
C: abundance of Corophium
S: abundance of snails (constant)
S (Snail abuindance)
Bifurcation diagram: lines represent changes in the system’s dynamics
B2
(speed of the snail response)
S (Snail abuindance)
Bifurcation diagram: lines represent changes in the system’s dynamics
I will start to
show the
dynamics
here:
B2
(speed of the snail response)
Corophium abundance)
when s=0.
phase plane: diatoms and corophium oscillate.
Diatom abundance
as snail abundance increases, oscillations first
disappear, and then populations crash
Corophium abundance)
.
S=1
Limit cycle:
Diatom abundance
S=1
Corophium abundance)
as snail abundance increases, oscillations first
disappear, and then populations crash.
Spiral to an equilibrium:
Limit cycle:
S=1
Diatom abundance
S=80
Corophium abundance)
as snail abundance increases, oscillations first
disappear, and then populations crash.
Limit cycle: S=1
Spiral to an equilibrium:
Diatom abundance
S=80
Corophium abundance)
as snail abundance increases, oscillations first
disappear, and then populations crash.
Spiral to an equilibrium:
S=81
S=80
Diatom abundance
Equilibrium disappear
S (Snail abuindance)
Bifurcation diagram: lines represent changes in the system’s dynamics
I will now
start to show
the
dynamics
here:
B2
(speed of the snail response)
Corophium abundance)
when s=0.
phase plane: diatoms and corophium oscillate.
Diatom abundance
Corophium abundance)
as snail abundance increases, oscillations first disappear, and
then populations crash.
Diatom abundance
Corophium abundance)
as snail abundance increases, oscillations first
disappear, and then populations crash.
Limit cycle:
S=1
Spiral to an equilibrium:
S=81
Diatom abundance
Corophium abundance)
as snail abundance increases, oscillations first
disappear, and then populations crash.
Limit cycle:
S=1
Spiral to an equilibrium:
S=81
Diatom abundance
Corophium abundance)
as snail abundance increases, oscillations first disappear, and then populations crash.
S is 200
Diatom abundance
SUMMARY AND NEXT STEPS
1. As snail abundance increases from zero, the
Corophium-diatom dynamics shift from
oscillating, to monotonic, to extinction of
Corophium and then diatoms.
2. The model will be extended to include snail
dynamics, and then simplified by assuming
diatom growth is rapid.
Thank you