Instabilidades de Turing e
Competição Aparente em Ambientes Fragmentados
Marcus A.M. de Aguiar
Lucas Fernandes
IFGW - Unicamp
Padrões de densidade em regiões homogêneas
Vegetation in arid ecosystems
Wolfram’s Cellular Automata
After 250 iterations
Complex Patterns can emerge out of simple interactions
between neighboring cells on a homogeneous environment
Cellular automata provide examples where direct interactions occur
Another Mechanism:
Reaction-Diffusion Systems and Turing Patterns
1912-1954
Summary
1. Turing Patterns in Homogeneous Environments
2. Turing Patterns in Networks of Fragments
3. Turing Patterns and Apparent Competition in Networks
1. Turing Patterns in Homogeneous Environments
Reaction
Populations of Preys (P) and Predators (Q)
dP
  P   PQ
dt
Lotka-Volterra Model
dQ
  Q   P Q
dt
,,c and  = constant coefficients
The Mimura-Murray model
On a diffusive prey-predator model which exhibits patchiness
M. Mimura and J.D. Murray
J. Ther. Biol. (1978) 75, 249-262
 a  bP  P 2

 
Q P
dt 
c

dP
dQ
dt
  P  (1  eQ )  Q
a  35,
b  16,
c  9,
e  2/5
Understanding the equation for the preys
The growth rate per capta is
1 dP
P dt

a  bP  P
2
Q
c
logistic growth
decrease because of predators
Allee effect
interspecies competition
P
Understanding the equation for the predators
The growth rate per capta is
1 dQ
  (1  eQ )  P
Q dt
logistic growth
Q
increase because of preys
mortality rate + interspecies competition
Diffusion
Add a new ingredient: space
P(x,t) = population of preys at spatial position x at time t
Q(x,t) = population of predators at spatial position x at time t
Add a new interaction: migration
The populations at x change by sending a fraction of its individuals to
neighboring sites and receiving a fraction of individuals from the same sites.
The math of diffusion
k-1
k
k+1
P(k-1) = 5
P(k)
=7
P(k+1) = 8
P(k) – P(k-1) = 2
P (k )
P(k+1) – P(k) = 1
 P ( k  1)  P ( k )    P ( k )  P ( k  1)  
P ( k  1)  P ( k  1)  2 P ( k )
A gradient in P(k) is not enough: diffusion requires a gradient of the gradient
 P
2
d
x
2
back
A uniform gradient is not enough
k-1
k
k+1
P(k-1) = 5
P(k)
=7
P(k+1) = 9
P(k) – P(k-1) = 2
P (k )
P(k+1) – P(k) = 2
 P ( k  1)  P ( k )    P ( k )  P ( k  1)   0
The new equations describing the dynamics of
Preys (P) and Predators (Q) become
2
 a  bP  P 2

 P
 
 Q P  dP
2
t
c

x


P
diffusion
local interactions
Q
t

 P  (1  eQ )  Q
 Q
2

dQ
x
2
diffusion coefficients
Partial Differential Equations
Can we understand these equations?
Are there simple solutions?
1 - Look for solutions that are uniform in space, i.e., situations where
the populations are the same at all points in space. In this case there
is no diffusion!
2
2


P
a  bP  P
 P
 
 Q P  dP
2
t
c

x


Q
t

 P  (1  eQ )  Q
 Q
2

dQ
x
2
2 - Look for solutions that are constant in time:
 a  bP  P 2

 
Q P
t
c


P
Q
t

 P  (1  eQ )  Q
Simplified equations can be solved:
 a  bP  P 2

Q P  0

c


 P  (1  eQ )  Q
a  35,
b  16,
c  9,
0
e  2/5
Two solutions:
1 – extinction
Q=P=0
2 – coexistence Q0 = 10 P0 = 5.
Are they stable?
J. Theor. Biol. 1978
u=P=prey
v=Q=pred
Patterns of preys and predators emerging on a homogeneous
environment.
Preys distributed on patches.
Predators everywhere, but with larger populations where the preys live.
2. Turing Patterns in Networks of Fragments
Predator-Prey systems on a Network:
Two main difficulties:
1 – describe diffusion in the network
2 – do the stability analysis
Describe the network:
1 - labels the nodes from 1 to N in order of decreasing number of
connections.
2 – Define the N x N adjancency matrix
Aij = 1 if nodes i and j are connected
Aij = 0 if nodes i and j are not connected
3 – ki = number of connections of node i
Examples:
4
3
5
5
1
0

1

A  0

0
1

2
3
1
0
0
0
1
0
1
0
1
0
1
0
0
0
0
1

0

0

0
0 
1
4
0

1

A  1

1
1

2
1
1
1
0
1
0
1
0
0
0
0
0
0
0
0
1

0

0

0
0 
Model diffusion:
Diffusion matrix
Diffusion of u at node i
L ij  Aij  k i  ij
N
L
N
ij
uj 
j 1
where ui is the population at node i

ji
Aij u j  k i u i
Example 1:
5
0

1

A  0

0
1

1
2
1
0
0
0
1
0
1
0
1
0
1
0
0
0
0
3
1

0

0

0
0 
4
 2

1

L 0

 0
 1

1
0
0
2
1
0
1
2
1
0
1
1
0
0
0
1 

0

0 

0 
 1 
 u1

u
 2
u   u3

 u4
u
 5








N
For node 2:
L
2 j
u j  u1  u 3  2 u 2
j 1
diffusion
Example 2:
4
3
5
1
0

1

A  1

1
1

1
1
1
0
1
0
1
0
0
0
0
0
0
0
0
2
 4

1

L 1

 1
 1

1

0

0

0
0 
1
1
1
2
1
0
1
2
0
0
0
1
0
0
0
N
For node 1:
L
1j
j 1
u j  u 2  u 3  u 4  u 5  4 u1
1 

0

0 

0 
 1 
 u1

u
 2
u   u3

 u4
u
 5








New Notation:
u = prey = P
v = predator = Q
N
 a  bu i  cu i 2

 
 v i  u i    L ij u j
dt
c
j 1


du i
dv i
dt
N
  u i  (1  dv i )  v i    L ij v j
each node is a fragment,
a local community
predator
v
j 1
For zero diffusion we are back to the same equations, for which there
is a homogeneous solution: each community has the same number
of preys and predators. We find
ui = u0 = 5 and vi = v0 = 10 for all nodes
prey
u
turn on diffusion:
0.12 15.6
Movies
Hysteresis
N
A
  ( u
i 1
i
2
2
 u )  ( v i  v ) 
3. Turing Patterns and Apparent Competition in Networks
N
 a  bu i  cu i 2

 
 v i  u i   u i y i    L ij u j
dt
c
j 1


du i
dv i
dt
N
  u i  (1  dv i )  v i     L ij v j
j 1
N
 a  bx i  cx i 2

 
 y i  x i    L ij x j
dt
c
j 1


dx i
predators
v
preys
u

dy i
dt
N
  x i  (1  dy i )  y i  f  u i y i    L ij y j
j 1
y
x
Results
Parameters: =0.12
=20.0
BA network with N=1000
Dynamics for first prey, u
Dynamics for second prey, x
Dynamics for u-x
=0.05
f=0.5
Effect of Coupling
Three nested predator-prey pairs in each node
predators
v
y


preys
u
x

z
Typical patterns:
sites with v-u and z-w and low values of y-x
sites with y-x and low values of v-u and z-w
few sites with all species in equal proportions
w
Four predator-prey pairs in each node
predators
v
y

z

r


preys
u
x

w
s
Typical patterns:
sites with v-u and z-w and low values of y-x and r-s
sites with y-x and r-s and low values of v-u and z-w
few sites with all species in equal proportions
Conclusions
• on a homogeneous environment, density patterns
can be generated dynamically, independent of
intrinsic differences.
• on a fragmented environment with identical patches,
abundance distributions can be different: there will
be two types of patches: with high abundance and
with low abundance.
• if more pairs of antagonistic species interact in each
patch, strong effects of apparent competition can
also be dinamically generated. There will be four
types of patches:
-
high v and u with low y and x
high y and x with low v and u
low v, u, y and x.
high v, u, y and x.