x
C
x2(r0)
?
stable
xcr
x1(r0)
0
unstable
stable
0
unstable
rcr
1
r0
Figure 1. Shown is a generic plot of the bifurcating continuum  for equation (23)
when    and, consequently, a backward (unstable) bifurcation occurs at the point
(0  ) = (1 0) The question mark indicates that although the positive equilibria on the
upper branch 2 (0 ) are (locally asymptotically) stable near the saddle-node bifurcation
point (   ), they can, depending on model parameter values, destabilize further
along the continuum .
x
x
(A)
20
16
15
12
10
8
5
4
0
(B)
0
0
1
2
4
r0
8
10
12
0.4
r0
1.0
1.4
Figure 2. Sample bifurcation diagrams for equation (23) with  = 03 and  = 3.
A.  = 005 and  =  −  = 015  0 so that the bifurcation at 0 = −3 = 1 is
forward and stable.
B.  = 1 and  = −27  0 so that the bifurcation at 0 = −3 = 1 is backward
and unstable (dashed line).
25
x
x
(A)
(B)
16
20
15
12
10
8
5
4
0
0
1
2
4
r0*
0
6
8
10
10
0
0.3
r0*
1.0
1.5
x
u1
u1
0.5
0.6
0.4
0.3
u2
u2
0.1
0
0
1
2
4
0.2
r0*
0
6
8
10
10
0
0.3
r0*
1.0
Figure 3. Sample bifurcation diagrams for equations (24) with  = 03  = 3 1 =
3 2 = 1 1 = 1 and 2 = 3 ¡
¢
A. 0 = 01 and ∗ = 3 1 − −23 10 ≈ 0146  0 so that the bifurcation at
0∗ = −3 = 1 is forward and
¡ stable. ¢
B.  = 1 and ∗ = 3 1 − 10−23 10 ≈ −12403  0 so that the bifurcation at
0∗ = −3 = 1 is backward and unstable (dashed lines).
26
1.5
(A)
(B)
10.0
4.8
x
7.5
2.6
x
5.0
2.4
2.5
1.2
u1
u2
u2
0
0
50
150
150
200
t
u1
0
0
50
150
150
200
150
150
200
t
1.0
4.8
0.8
2.6
0.6
u1
2.4
0.4
u2
x
1.2
0.1
u2
0
0
x
u1
50
150
150
200
t
0
0
50
Figure 4. Shown are sample orbits for equations (24) with the same parameter values
used in Figure 3B when a backward bifurcation creates an interval of 0∗  1 for which
there is a strong Allee effect.
A. For 0∗ = −3 = 04 the bifurcation diagram Figure 3B shows a stable extinction
equilibrium and a stable positive equilibrium. The upper graph in column A shows
plots of the solution with initial conditions  = 235 1 = 2 = 1 and that tends to the
positive equilibrium. The lower graph shows plots of the solution with initial conditions
 = 233 1 = 2 = 1 and that tends to the extinction equilibrium ( ̂) = (0 0̂).
B. For 0∗ = −3 = 09 the bifurcation diagram Figure 3B shows a stable extinction
equilibrium and a stable 2-cycle. The upper graph in column B shows plots are of the
solution with initial conditions  = 235 1 = 2 = 1 and that tends to the positive
2-cycle. The lower graph in column B shows plots of the solution with initial conditions
 = 035 1 = 2 = 1 and that tends to the extinction equilibrium ( ̂) = (0 0̂).
27
t