Parameter values for the Fig. 2: D = 0.5, S = 19, ri = 1, mi = 0.5, νi and qi values are given below. The values of Ki and θi are chosen at random between 0.1 to ...
Oikos
OIK-02322
Chakraborty, S., A, R. and Dutta, P. S. 2015. Toxic phytoplankton as a keystone species in aquatic ecosystems: stable coexistence to biodiversity. – Oikos doi: 10.1111/oik.02322
Appendix 1: Parameter values used for our numerical simulations Table A1. Parameter values for the Fig. 1: D = 0.5, S = 19, ri = 1, mi = 0.5, νi and qi values are given below. The values of Ki and θi are chosen at random between 0.1 to 0.84 and 0 to 0.1 respectively. Initial conditions (t = 0): R = S, Qi = qi and Ni = 0.01.
Species index i
1
2
3
4
5
6
7
8
νi
1.76
2.25
1.61
1.8
1.9
2.1
2.05
1.95
qi
1.40
1.97
1.38
1.5
1.55
1.7
1.8
1.65
Table A2. Parameter values for the Fig. 2: D = 0.5, S = 19, ri = 1, mi = 0.5, νi and qi values are given below. The values of Ki and θi are chosen at random between 0.1 to 0.84 and 0 to 0.1 respectively. Initial conditions (t = 0): R = S, Qi = qi and Ni = 0.01. Species index i
1
2
3
4
5
6
7
8
νi
1.76
2.25
1.61
1.8
1.9
2.1
2.05
1.95
qi
1.40
1.97
1.38
1.5
1.55
1.7
1.8
1.65
Ki
0.8250
0.4248
0.1822
0.2910
0.4025
0.5402
0.2940
0.5461
θi
0
0.0711
0.0222
0.0117
0.0297
0.0319
0.0424
0.0508
1
Table A3. Parameter values for the Fig. 3 : ri = 1, mi = D, the values of νi, qi and Ki are given below. The values of D and S are varied from 0 to 0.6 and 0 to 50 respectively. Initial conditions (t = 0): R = S, Qi = qi and Ni = 0.01.
Species index i
1
2
νi
1.76
2.25
qi
1.40
1.97
Ki
0.43
0.21
2
Table A4. Parameter values for the Fig. 6: D = 0.5, ri = 1, mi = 0.5, νi and qi values are given below. The value of S is varied from 0 to 200. The values of Ki and θi are chosen at random between 0.1 to 0.84 and 0 to 0.1 respectively. Initial conditions (t = 0): R = S, Qi = qi and Ni = 0.01. For red curve Species index i
1
2
3
4
5
6
νi
1.76
1.8
1.9
2.1
2.05
1.95
qi
1.40
1.5
1.55
1.7
1.8
1.65
Ki
0.7142
0.1982
0.5954
0.1585
0.1981
0.3221
θi
0
0.0900
0.0655
0.0639
0.0739
0.0393
For blue curve Species index i
1
2
3
4
5
6
7
8
νi
1.76
2.25
1.61
1.8
1.9
2.1
2.05
1.95
qi
1.40
1.97
1.38
1.5
1.55
1.7
1.8
1.65
Ki
0.7097
0.3833
0.3617
0.3446
0.6198
0.3841
0.1230
0.4379
θi
0
0.0279
0.0336
0.0052
0.0540
0.0381
0.0998
0.0530
3
Table A5. Parameter values for the Fig. 7: D = 0.5, S = 19, ri = 1, mi = 0.5, νi and qi values are given below. The values of Ki and θi are chosen at random between 0.1 to 0.84 and 0 to 0.1 respectively and the value of θc is varied from 0 to 200. Initial conditions (t = 0): R = S, Qi = qi and Ni = 0.01.
Species index i
1
2
3
4
5
6
7
8
νi
1.76
2.25
1.61
1.8
1.9
2.1
2.05
1.95
qi
1.40
1.97
1.38
1.5
1.55
1.7
1.8
1.65
Ki
0.6845
0.2749
0.2734
0.4439
0.5583
0.5553
0.1907
0.1916
θi
0
0.0284
0.0736
0.0411
0.0829
0.0935
0.0399
0.0052
4
Appendix 2: Two species–one resource system with allelopathy In the absence of allelopathy (i.e. θi = 0), the competition model given by Eq. 1 (see the main text) has only one equilibrium solution. It goes a long way back to the “competitive exclusion principle” by G. F. Gause (1934). ∗ The winner of the competition depends upon the resource requirement Rji of
the i-th species for the j-th resource. The specific i-th species, which has the lowest requirement for the j-th resource will be the winner of the competition. Now, if θi �= 0 for i = 2, . . . , n, the result is remarkably different. Due to the additional nonlinearity in Eq. 1d there can be n-stable equilibrium solution for the species. The allelochemical interaction promotes stable coexistence of nspecies only on single resource. Assuming only two species competing for a single resource, now we will describe the analytical result. Only for two species, the Eq. 1 are described below: dR dt dQ1 dt dQ2 dt dN1 dt dN2 dt
ν1 R ν2 R N1 − N2 , K1 + R K2 + R ν1 R q1 = − r1 (1 − )Q1 , K1 + R Q1 ν2 R q2 = − r2 (1 − )Q2 , K2 + R Q2 q1 = r1 (1 − )N1 − m1 N1 , Q1 q2 = r2 (1 − )N2 − m2 N2 − θ2 N12 N22 . Q2 = D(S − R) −
5
(2a) (2b) (2c) (2d) (2e)
When θ2 = θ �= 0, the model given by Eq. 2 has the following possible equilibria: 1) the phy toplankton free equilibrium: ER = (RR, QR1 , QR 2 , 0, 0); 2) the toxic monoculture equilibrium: ET = (RT , QT 1 , QT2 , N1T, 0); 3) the T NT NT non-toxic monoculture equilibrium: EN T = (RN T , QN 1 , Q2 , 0, N2 ); and
4) the poly culture equilibrium EP = (R∗, Q∗ ), where both 1 , Q2 , N 1 , Nthe 2 ∗
∗
∗
toxic and the non-toxic species can coexist. The polyculture equilibrium EP = (R∗ , Q∗1 , Q∗2 , N1∗ , N2∗ ) is given by the following equations: r1 q 1 m 1 K 1 , ν1 (r1 − m1 ) − r1 q1 m1 r1 q1 Q∗1 = , r1 − m1 (ν2 + r2 q2 )R∗ + r2 q2 K2 , Q∗2 = r2 (K2 + R∗ )
(3b)
N2∗ + C1 N1∗ = C2 ,
(3d)
R∗ =
(N1∗ )2 N2∗ =
C1 =
C3 , where θ
(3a)
(3c)
(3e)
ν1 (K2 + R∗ ) D(S − R∗ )(K2 + R∗ ) q2 , C = and C = r (1 − ) − m2 . 2 3 2 ν2 (K1 + R∗ ) ν2 R ∗ Q∗2
Solving the Eq. 3d and 3e one can get nonzero positive solutions for N1∗ and N2∗ depending upon the value of the allelopathy parameter θ. There is a threshold θt (a critical value) for the allelopathy parameter θ; if the value of θ < θt there does not exist any positive solution, if the value of θ = θt , there is only one solution and for θ > θt , there exists two fixed
points (marked by arrows in Fig. A1). One of the fixed points is unstable (EPu ) and another one is stable (EPs ). The stable fixed point (EPs ) is giving 6
us the desired stable coexistence of two species only on one resource. One can generalize the above calculations for n-species. 1.4 1.2 1.0
EPu
N2
0.8 Θ
Θt
0.6 Θ
Θt
0.4 Θ
0.0 1.0
Θt
EPs
0.2
1.5
2.0
2.5
3.0
3.5
N1
Figure A1. Existence of nonzero positive equilibrium for both the toxic and the non-toxic species. The straight line is given by Eq. 3d and curves are given by Eq. 3e for different values of the allelopathy parameter θ. For certain threshold θ (= θt), a unique equilibrium is obtained by the intersection of the curve and the straight line. Below this threshold no equilibrium exists whereas above that two distinct non zero equilibrium points exist which are marked with arrows. The fixed point in the lower right corner is stable (EPs ) and the fixed point in the upper left corner is unstable (EPu ).
The parameter values we used for the above figure are S = 10, r1 = r2 = 1, D = m1 = m2 = 0.5 and other values are given in the below table with the allelopathy parameter θt = 1 × 10−3 (approx).
Species no. νi Ki qi
i=1 2.2 0.96 1.4
7
i=2 2.22 0.28 1.88
References Gause, G. F. 1934. The struggle for existence. - Hafner, NY.
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