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Differential dispersion in planktonic food chains with constant inputs
Ecological Modelling, 15 (1982) 75-78 Elsevier Scientific Publishing Company, Amsterdam--Printed in The Netherlands
75
Short Communication DIFFERENTIAL DISPERSION WITH CONSTANT INPUTS
IN P L A N K T O N I C
FOOD
CHAINS
RICHARD A. PARKER Departments of Zoology and Computer Science, Washington State University, Pullman, WA 99164 (U.S.A.) (Accepted for publication August 1981) Parker, R.A., 1982. Differential dispersion in planktonic food chains with constant inputs. Ecol. Modelling, 15: 75-78. INTRODUCTION Hastings (1978) recently derived sufficient conditions for global stability in n-species Lotka-Volterra (LV) systems with diffusion, both in continuous environments having no flux at the boundaries and environments consisting of discrete patches. His work extended the two-species case explored by Rothe (1976), Conway and Smoller (1977), and Leung (1978). Earlier, Levin and Segel (1976) concerned themselves with the destabilizing influence of diffusion on plankton populations near equilibrium, when the phytoplankton were affected by an autocatalytic term. Okubo (1978) examined the effects of differential dispersion and concluded that it was premature, at present, to assert that diffusion-induced instability played a role in nature. Results reported here also detail the consequences of differential dispersion on interacting populations, but allow boundary flux, specifically constant inputs analogous to those observed at times in marine upwelling systems or long, narrow reservoirs.
MODEL AND RESULTS Consider a two-species system, usually nonlinear, described by
=L(.,..2) in which u 1 represents phytoplankton and u 2 zooplankton. Parker (1978) 0304-3800/82/0000-0000/$02.75 © 1982 Elsevier Scientific Publishing Company
76 presented a steady-state solution
= e x p [ r e ] w(O) say to the partial differential equation in time and space Wt = D w y y
-- VWy "4-Aw
where w is a vector of variables linearized with respect to equilibrium u*, and w(0) is a constant input vector at the boundary Y = 0. A is the matrix of coefficients obtained from linearizing the original nonlinear system in u, V is the water velocity in one spatial dimension Y, and D = diag(d I, d2) represents fixed dispersion coefficients. Note that only the positive square root leads to the solution w(0) necessary at Y = G¢ when A = 0. Now let B = (( VD - 1.)2 _ 4D - IA), that is B = [ ( V / d ' ) 2 -- 4 a , , / d ,
l -4a2,/d2
--4al2/d , (Z/d2) 2 -
Then if ( b l l - - b 2 2 ) 2 < - 4 b l 2 b 2 1 , B has complex eigenvalues ~, C is a corresponding matrix of eigenvectors, and S = B I/2 = C diag[~ 1/2] C -I s=[R+((b"-b22)/2}G/g
[ b2,G/g
b'2G/g 1 R-- {(b,,-b22)/Z}G/g
Here ~ = r ± gi; r = ( b~ + b22)/2; g = ((b~b22 - bt2b2~ ) - ¼( b~ + b22)2}'/2; ~,/2 = R +-- Gi;R = [½((b~b2z - b~2b2~) 1/2 + (b~ + b22)/2}l'/2;G = [½((bi~bzz-b~2bz~)~/2-(b|~ +bz2)/2}]1/2; G / g = ½ R . It follows that the elements of E are
±[V/d,-s,, 2 __s21
--Si2 ] V/dR _ s221
For the usual LV model, all ---a22 = 0 , al2 < 0 , a21 > 0 . In this case, (ell + e 2 2 ) < 0 and (elle22--el2e21)>0, since ell < 0 , e22 *(0, el2 < 0 , e21 > 0 . * Hence the real parts of the eigenvalues of E are negative, and the system is always stable. In fact, stability is assured if all and a22 ~ 0, d I and d 2 > 0. If both d~ and d 2 --, 0 B
0 E~ a2j/V
al2/V 0
77
This result corresponds to the steady-state solution w(Y) = exp[ TA] w(0) where Y / V is the travel time T from the point of entrance to a fixed point Y. Unfortunately, asymptotic stability cannot be guaranteed for a linearized LV system with three or more speci,es unless all dispersion coefficients are equal (d i = d say). T h e n
"Oi~----l[Vd-' - ((Vd-')2-4~/i) 1/2] is an eigenvalue of E given Yi, an eigenvalue of d - ~ A. If Yi is real and negative, so is ~,; if Yi = a + fli and a is zero or negative, then the real part of ~, is negative. When the dispersion coefficients are not equal, suitable constraints on A can be introduced to maintain stability. For example, consider the straight chain p r e d a t o r - p r e y system UI ' = (kll --k12u2) U| U 2 ' : (k21u I - k 2 3 u 3 ) u2
u,,' = ( k,,~.,_,)u._ 1 -- k n,,) u. and correspondingly
0 k21u ~ A =
0 0
--kl2U ~ 0 k32u~
0
...
0
--k23u ~ 0
k.~._l~u*
- k ~ _ 1)nu*n-- I 0
Here the real parts of ,/ are < 0; however, if n is odd, the system decays to n - 1 species. The collapse is associated with two conflicting parenthetical expressions of per capita birth and death rates defining equilibrium population densities, a mathematical peculiarity of long standing (May, 1973). This problem is easily remedied by making k~l a strictly decreasing function of u 1 (see Harrison, 1979), or k,n a strictly increasing function of u,. CONCLUSION
These results suggest that instability is rare in naturally occurring planktonic systems subject to turbulent transport and constant inputs, even though the organisms may disperse at different rates. Furthermore, spatially
78
oscillating populations should not be expected over extended distances from the input source. REFERENCES Conway, E. and Smoller, J., 1977. Diffusion and the predator-prey interaction. SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 33: 673-686. Harrison, G.W., 1979. Global stability of food chains. Am. Nat., 114: 455-457. Hastings, A., 1978. Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol., 6: 163-168. Leung, A., 1978. Limiting behavior for a prey-predator model with diffusion and crowding effects. J. Math. Biol., 6: 87-93. Levin, S.A. and Segel, L.A., 1976. An hypothesis to explain the origin of planktonic patchiness. Nature, 259: 659. May, R.M., 1973. Stability and complexity in model ecosystems. Princeton Univ. Press, Princeton, NJ, 235 pp. Okubo, A., 1978. Horizontal dispersion and critical scales for phytoplankton patches. In: J.H. Steele (Editor), Spatial Pattern in Plankton Communities. Plenum, New York, pp. 21-42. Parker, R.A., 1978. Spatial patterns in a nutrient-plankton model. Ecol. Modelling, 4: 361-370. Rothe, R., 1976. Convergence to the equilibrium state in the Volterra-Lotka diffusion equations. J. Math. Biol., 3: 319-324.
75
Short Communication DIFFERENTIAL DISPERSION WITH CONSTANT INPUTS
IN P L A N K T O N I C
FOOD
CHAINS
RICHARD A. PARKER Departments of Zoology and Computer Science, Washington State University, Pullman, WA 99164 (U.S.A.) (Accepted for publication August 1981) Parker, R.A., 1982. Differential dispersion in planktonic food chains with constant inputs. Ecol. Modelling, 15: 75-78. INTRODUCTION Hastings (1978) recently derived sufficient conditions for global stability in n-species Lotka-Volterra (LV) systems with diffusion, both in continuous environments having no flux at the boundaries and environments consisting of discrete patches. His work extended the two-species case explored by Rothe (1976), Conway and Smoller (1977), and Leung (1978). Earlier, Levin and Segel (1976) concerned themselves with the destabilizing influence of diffusion on plankton populations near equilibrium, when the phytoplankton were affected by an autocatalytic term. Okubo (1978) examined the effects of differential dispersion and concluded that it was premature, at present, to assert that diffusion-induced instability played a role in nature. Results reported here also detail the consequences of differential dispersion on interacting populations, but allow boundary flux, specifically constant inputs analogous to those observed at times in marine upwelling systems or long, narrow reservoirs.
MODEL AND RESULTS Consider a two-species system, usually nonlinear, described by
=L(.,..2) in which u 1 represents phytoplankton and u 2 zooplankton. Parker (1978) 0304-3800/82/0000-0000/$02.75 © 1982 Elsevier Scientific Publishing Company
76 presented a steady-state solution
= e x p [ r e ] w(O) say to the partial differential equation in time and space Wt = D w y y
-- VWy "4-Aw
where w is a vector of variables linearized with respect to equilibrium u*, and w(0) is a constant input vector at the boundary Y = 0. A is the matrix of coefficients obtained from linearizing the original nonlinear system in u, V is the water velocity in one spatial dimension Y, and D = diag(d I, d2) represents fixed dispersion coefficients. Note that only the positive square root leads to the solution w(0) necessary at Y = G¢ when A = 0. Now let B = (( VD - 1.)2 _ 4D - IA), that is B = [ ( V / d ' ) 2 -- 4 a , , / d ,
l -4a2,/d2
--4al2/d , (Z/d2) 2 -
Then if ( b l l - - b 2 2 ) 2 < - 4 b l 2 b 2 1 , B has complex eigenvalues ~, C is a corresponding matrix of eigenvectors, and S = B I/2 = C diag[~ 1/2] C -I s=[R+((b"-b22)/2}G/g
[ b2,G/g
b'2G/g 1 R-- {(b,,-b22)/Z}G/g
Here ~ = r ± gi; r = ( b~ + b22)/2; g = ((b~b22 - bt2b2~ ) - ¼( b~ + b22)2}'/2; ~,/2 = R +-- Gi;R = [½((b~b2z - b~2b2~) 1/2 + (b~ + b22)/2}l'/2;G = [½((bi~bzz-b~2bz~)~/2-(b|~ +bz2)/2}]1/2; G / g = ½ R . It follows that the elements of E are
±[V/d,-s,, 2 __s21
--Si2 ] V/dR _ s221
For the usual LV model, all ---a22 = 0 , al2 < 0 , a21 > 0 . In this case, (ell + e 2 2 ) < 0 and (elle22--el2e21)>0, since ell < 0 , e22 *(0, el2 < 0 , e21 > 0 . * Hence the real parts of the eigenvalues of E are negative, and the system is always stable. In fact, stability is assured if all and a22 ~ 0, d I and d 2 > 0. If both d~ and d 2 --, 0 B
0 E~ a2j/V
al2/V 0
77
This result corresponds to the steady-state solution w(Y) = exp[ TA] w(0) where Y / V is the travel time T from the point of entrance to a fixed point Y. Unfortunately, asymptotic stability cannot be guaranteed for a linearized LV system with three or more speci,es unless all dispersion coefficients are equal (d i = d say). T h e n
"Oi~----l[Vd-' - ((Vd-')2-4~/i) 1/2] is an eigenvalue of E given Yi, an eigenvalue of d - ~ A. If Yi is real and negative, so is ~,; if Yi = a + fli and a is zero or negative, then the real part of ~, is negative. When the dispersion coefficients are not equal, suitable constraints on A can be introduced to maintain stability. For example, consider the straight chain p r e d a t o r - p r e y system UI ' = (kll --k12u2) U| U 2 ' : (k21u I - k 2 3 u 3 ) u2
u,,' = ( k,,~.,_,)u._ 1 -- k n,,) u. and correspondingly
0 k21u ~ A =
0 0
--kl2U ~ 0 k32u~
0
...
0
--k23u ~ 0
k.~._l~u*
- k ~ _ 1)nu*n-- I 0
Here the real parts of ,/ are < 0; however, if n is odd, the system decays to n - 1 species. The collapse is associated with two conflicting parenthetical expressions of per capita birth and death rates defining equilibrium population densities, a mathematical peculiarity of long standing (May, 1973). This problem is easily remedied by making k~l a strictly decreasing function of u 1 (see Harrison, 1979), or k,n a strictly increasing function of u,. CONCLUSION
These results suggest that instability is rare in naturally occurring planktonic systems subject to turbulent transport and constant inputs, even though the organisms may disperse at different rates. Furthermore, spatially
78
oscillating populations should not be expected over extended distances from the input source. REFERENCES Conway, E. and Smoller, J., 1977. Diffusion and the predator-prey interaction. SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 33: 673-686. Harrison, G.W., 1979. Global stability of food chains. Am. Nat., 114: 455-457. Hastings, A., 1978. Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol., 6: 163-168. Leung, A., 1978. Limiting behavior for a prey-predator model with diffusion and crowding effects. J. Math. Biol., 6: 87-93. Levin, S.A. and Segel, L.A., 1976. An hypothesis to explain the origin of planktonic patchiness. Nature, 259: 659. May, R.M., 1973. Stability and complexity in model ecosystems. Princeton Univ. Press, Princeton, NJ, 235 pp. Okubo, A., 1978. Horizontal dispersion and critical scales for phytoplankton patches. In: J.H. Steele (Editor), Spatial Pattern in Plankton Communities. Plenum, New York, pp. 21-42. Parker, R.A., 1978. Spatial patterns in a nutrient-plankton model. Ecol. Modelling, 4: 361-370. Rothe, R., 1976. Convergence to the equilibrium state in the Volterra-Lotka diffusion equations. J. Math. Biol., 3: 319-324.