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Spatial pattern and the mechanism of population regulation
J. theor. Biol. (1976) 59, 361-370
Spatial Pattern and the Mechanism of Population Regulation W. S. C. GURNEY AND R. M. NISBJS Department of Applied Physics, University of Strathcijde, Glasgow, Scotland (Received 31 July 1975) It has been suggested that the saturation density of many populations is adjusted to match environmental conditions by indirect controls rather than by direct physiological response to lack of some critical resource. This hypothesis is widely regarded as untestable since the observable effects of direct and indirect regulation have been held to be indistinguishable. In this paper we show that the way in which the density of a consumer population competing for a single, spatially heterogeneous, limiting resource, “tracks” variations in its environment is characteristically dependent upon the nature of the density limiting mechanism.
1. Introduction Wynne-Edwards (1962) has contended that the saturation density of animal populations is adjusted to match optimally their environmental conditions by indirect (e.g. behavioural) controls rather than by direct physiological response to lack of a particular resource. It has been widely believed (see, for example, Ricklefs, 1973) that this hypothesis is almost untestable since the observable effects of direct and indirect regulation are essentially indistinguishable. In this paper we show that the way in which the density of a consumer population competing for a single, spatially heterogeneous, limiting resource “tracks” the underlying variations in its environment is characteristically dependent upon the nature of the density limiting mechanism. The problem of the effect of intra-species competition on the way in which population density “tracks” a spatially heterogeneous environment has been considered by Roughgarden (1974) using a “logistic” model analogous to his (1971) model of interspecies competition for a spectrum of resources. His most striking conclusion is that increasing the size of the individual resource capture zone increases the representation of fine scale environmental variations in the pattern of population abundance. This conclusion emerges largely because the Roughgarden model describes only the negative (i.e. 361
362
W.
S.
C.
GURNCY
AND
R.
M.
NISBET
competitive) effects of a finite resource capture zone and neglects the beneficial effects of resource averaging. We have examined the “tracking problem” using a two species model which is a generalization of MacArthur’s (1969, 1970) model of interspecific competition. We consider consumers of density N(r, t) feeding on a single resource of density R(r, t). Since we wish to describe a situation in which a consumer at r feeds on resource in a locality around r, we must be careful about the meaning we ascribe to consumer density. For most plants, and for non-territorial animal species the interpretation is reasonably clear, but in the case of an animal which gathers food over a large but relatively unchanging area we must be careful not to confuse “dispersal motion” with “food gathering motion”. This difficulty, which is discussed in detail by Gurney & Nisbet (1975), may be resolved by regarding N(r, t) as a time average taken over a time short enough to preclude significant dispersal but long enough to average out food gathering trips. We now follow Roughgarden, and describe the geographical extent of the food gathering activities of a consumer at r by means of a normalized resource capture function I(r, r’), whose simplest interpretation is essentially that in a perfectly uniform environment a consumer at r would draw a proportion I(r, r’) dr’ of its total food intake from dr’. We now define the average resource R*(r) available to a consumer at r in a non-uniform environment as R*(r) = .d,, W, WW dr’, (1) and the average feeding load on resources at r as N*(r) = j I(r’, r)N(r’) dr’. allrl
(2)
We assume that resource at position r is replaced at a rate x(&r), R(r)) dependent only upon the local resource density and the underlying favourability pf the locality specified by d(r), and is consumed at a rate F@(r), N*(r)) which depends on the local resource density and the local average feeding load. For simplicity we now define a local net growth rate +Kr), R(r), N*(r)) = x(M), R(r)) -NW, N*(r)), and, making the assumption that the resource species is essentially immobile, write &r> = 7&#@>, R(r), N*(r)). (3) For ease of later interpretation we describe the consumer population at r by a net specific growth rate G(N(r), R*(r)) depending, in some arbitrary way on both the local consumer density N(r) and the average locally available resource R*(r). We envisage the dispersal of individual consumers from r as
SPATIAL
PATTERN
AND
taking place by “jumps” occurring probability P(r, r’) dr’ of landing very simple piece of book-keeping g(r) = G(N(r), R*(r))iV(r)
POPULATION
REGULATION
363
with an average frequency v and having a in dr’. Provided that P(r, r’) = P(r’, r) a now suffices to show that +v J P(r, r’){iV(r’)- N(r)} dr’. (4) all rl
2. Stationary States (A)
UNIFORM
ENVIRONMENTS
In our model the environmental heterogeneity is specified by the spatial variation of the favourability function 4(r). In the case of a homogeneous environment we set 4(r) = &,, and assume that our model has a spatially uniform steady state with R(r) = R, and N(r) = NO such? that G(N,, R,) = 0. d&l, R,, NCJ = 0; (5) In Appendix A, with the aid of the plausible assumption that both the resource capture function I(r, r’) and the jump probability distribution P(r, r’) are functions only of the interval Ir - r’l, we derive completely general conditions for the decay of fluctuations from any such steady state. However, if we consider only systems with “predator-prey” linking between consumer and resource (so that both aG/aR* and -d@N* are strictly positive) and exclude the case in which “autocatalytic” action causes the specific consumer growth rate to increase with consumer density, the stability of any stationary state is assured if, at equilibrium, (6)
and (7) These inequalities are simply a statement of the well known result that any system of interacting populations of the class described by equations (3) and (4) can be stabilized by the provision of sufficient density dependence in either trophic level. It is this result which is the basis of most doubts about the distinguishability of internal and external control. (B)
NON-UNIFORM
ENVIRONMENTS
We now wish to ask what stable population distribution will result from a given pattern of environmental heterogeneity. If the environmental fluctuations are sufficiently small for us to be able to write I * q& for all r, 4(r) = h + Xr> (8) t We note that since Jl(r, r’) dr’ E 1, R(r) = R. +-R*(r) = Rn, and N(r) = NO * N*(r) .= NO.
364
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then we can find an approximate analytic solution to this problem, by making the reasonable supposition that if (R,, NO) is the uniform steady state corresponding to 4(r) = &, then R(r) = R,+<(r) and
5 Q R, for all r
(9)
q 4 N, for all r.
W = No + r(r) (10) If, as before, we assume that I(r, r’) and P(r, r’) in fact depend only on the interval Ir-r’l then we see immediately that to a very good approximation the steady state distributions of resources and population are given by the simultaneous solutions of
and
($$W+(f$)
d,, W>Z@ - r’>dr’+ & air,P(r-r%(f) -WI dr’ = 0,
(12)
all partial derivatives being evaluated at (&,, R,, NJ. Equations (11) and (12) are hard to solve directly, but because we have chosen the functions I and P to be functions only of the interval r-r’, we can follow Roughgarden and simplify the process considerably by regarding each function of r as a superposition of sinusoidal components of different “spatial frequency” or wave-vector?. Thus we decompose some function 8(r) into a set of sinusoids O(k) eikar with wave-vector k and amplitude O(k), such that in a d-dimensional universe O(r) = & d j B(k)e ik.‘dk (13) 0 all k The orthogonality of these components enables us to obtain the spectrum e(k) corresponding to a particular O(r) rather easily as O(k) = a/ r 6(r) e-ik.r dr A particularly useful feature of the Fourier transformation equation (14) is that if C(r) = j A(r’)B(r - r’) dr’, all rl then C(k) = A(k)B(k).
(14) defined by (15) (16)
t The wave vector of a plane wave of wavelength 1 is a vector whose direction is normal to the wave front and whose magnitude is 274.l.
SPATIAL
PATTERN
AND
POPULATION
REGULATION
365
With this in mind we multiply both sides of equations (11) and (12) by e -ik.r and integrate over all r, making use of the convolution theorem [equations (15) and (16)] where appropriate. A little routine algebra now tells us that the component of environmental variation with wave-vector k and amplitude X(k) gives rise to a component of population density variation with the same wave-vector and an amplitude ij(k) given by (17) where T(k) =
(18)
($:) (-I$)w +
El-YOGI
The solution of our problem is now formally obtained by plugging (17) back into (13) and performing the requisite integration. However, it is usually more convenient to leave the solution in wave-vector space and deal directly with the “spatial transfer function” T(k). Although expression (18) for this function looks rather daunting we can relatively easily establish some of its more important qualitative properties. To simplify the discussion we shall assume that both the resource capture function Z(r-r’) and the jump probability distribution P(r - r’) are symmetrical Gaussians with -standard deviations (or and g,, respectively. This at once implies that their Fourier transforms are i(k) = exp ( -+o12k2), (19) and P(k) = exp (- +op2k2), (20) immediately enabling us to show that any stable system has a spatial transfer function which goes monotonically to zero as Ikl increases unless an i?G dG ’ _ ii~ + -5;. < “SF* I 0
(21)
-
in which case T(k) goes through a single maximum
before decreasing to zero,
3. Discussion The terms “internal” and “external” control are often somewhat loosely defined, but in the context of our model it is quite clear that external control of the consumer species is represented by @G/JR*), while the degree of internal control is measured by -(aG,@N),.. Since the right hand side of I .a. 24
366
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GURNEY
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M.
NISBET
inequality (21) must be positive for any non-trivial stable system, we see immediately that this condition for a maximum in the spatial transfer function tells us that the behaviour of a system in which the consumer level is internally regulated (- aG/&V large) is clearly and qualitatively different from the behaviour of a system whose consumer level is regulated externally, provided only that the consumer mobility is not so high as to mask the differences between the two control systems. An illustrative example may serve to illuminate this point. Consider two systems (models A and B) with resource dynamics &r> = R(r)
I - ifi - ajqr)~*(r)a ( > In model A we describe our consumer dynamics by A(r) = [/H*(r)
-YIN(r)
+ v!,P(r
- r’)[N(r’)
(22)
-N(r)]
dr’,
(23)
while in model B the consumers are assumed to behave as N(r) + vs P(r -r’)[N(r’)
- N(r)] dr’.
(24)
Both models are stable for any combination of parameters which produces strictly positive solutions for N(r) and R(r). The difference between them is that in model A control of the consumer level is wholly external (LJG/dN = 0) while in model B the consumers have a measure of internal control. Assuming that the resource capture function Z and the jump probability distribution P are both symmetrical gaussians with standard deviation rrr and gP respectively, equation (18) tells us that in terms of the normalized wave-vector k’ (=cr,k) the spatial transfer function for model A is
while that for model B is
T(k’)
=
--~~-.~
exp ( -IL’)
(26) +
Examples of the form of these two relations as a function of the modulus of normalized wave-vector are displayed in Fig. 1 for various values of the normalized consumer mobility oR[ = u,,/~,]. It is clear from this diagram that
SPATIAL
PATTERN
390
AND
POPULATION
REGULATION
367
-
0
0.6
I.2
Nortnolised
WOYO vector
2-4
3-O
24
30
modulus
(4
0
0.6 Normolised
I.2 wove
I3 vector
modulus
FIG. 1. Typical Spatial Transfer Functions (a) Model with external consumer control only; (b) Model with strong internal consumer control.
368
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if the consumer species is very mobile ((TV > 0,) then the two both have monotonically decreasing transfer functions, but in the low mobility case (ap < CT&the internally regulated consumers show a monotonically decreasing transfer function while those regulated wholly externally show a transfer function with a strong maximum. Lastly we note that only for a model with no consumer internal control and no consumer dispersal does increasing the size of the resource catchment (a& always increase the representation of fine scale environmental differences in the pattern of population density. If the consumers are at all mobile this is true only for wave-vectors below the transfer-function maximum, and if the consumers have either enough mobility or enough internal control to eliminate the maximum from the transfer function then no vestige of this effect remains. 4. C!onclusion We have shown that the relative degree to which environmental variations of differing wavelengths are represented in the equilibrium dispersion of a population supported by a single, spatially heterogeneous, renewable resource is characteristically dependent upon the nature of the mechanism by which the population is stabilized, provided only the population is not so mobile as to mask the relevant effects, and that the variations concerned are not so large as to render a transfer function description unappropriate. An internally controlled or highly mobile population will show a density distribution in which the relative representation of each wavelength of environmental variability decreases monotonically as the wavelength itself decreases. An externally controlled and relatively immobile species will show a density distribution in which some wavelength range is relatively over-represented.
APPENDIX Stability
A
of the Stationary State
In this appendix we analyse the stability of the steady-state solutions of (Al) fib) = +Kr>, R(r), N*(r)>, fi(r) = G(W), R*(r))N(r) +a;i,P(r, r’)(W) - N(r)} dr’. 642) In a homogeneousenvironment, these equations have a uniform steady state with 440, Ro, No) = Wo, R,) = 0. (A3)
SPATIAL
PAT.1.ERN
AND
Let t(r), q(r) be small perturbations
POPULATION
REGULATION
369
in R and N. Then to hrst order 644)
s P(r, r’){N(r’) i =No@+No(g)(*+ ,;,,
- N(r)} dr’,
(A5)
where all partial derivatives are evaluated at the steady state. Provided P(r, r’) depends only on (r--t’J, we can take the spatial Fourier transform of the above equations and obtain W)
Necessary and sufficient conditions for the stability of the system (A6)-(A7) may be derived in the normal manner (see e.g. May 1973, Appendix II) and turn out to be
8G + v(P-1) < 0,
&+“oz and
N h3G-N OaRaN
&iYG
--z OaN* aR*
m2
+vg+)>O.
(A8) W)
These conditions must be satisfied for all wave vectors k if the system (A4)-(A5) is to be unconditionally stable. We now note that for all k, P(k) < 1 and that for “predator-prey” linkage an -
and
aG
i3R’ ’ 0.
With these conditions it is clearly sufficient for stability that in addition a71
- < 0, aR
dG
- < 0, aN
(All)
i.e. the system can be controlled by the provision of sufficient density dependence in either the producer or the consumer dynamics. The inequalities (Al 1) preclude the existence of an inhomogeneous “dissipative” steady state of the type considered by Segal & Jackson (1972). The above proof is not valid for an inhomogeneous environment since the various partial derivatives in equations (A4) and (A5) are then functions of position and we cannot easily Fourier transform the equations. However for
3 70
\V.
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C’.
CiURNl.1
ANi)
R.
M.
NISRl:I’
the mildly inhomogeneous steady states analysed in the text, equations (A4)-(A5) are first-order approximations to the perturbation equations at any point r and we conclude that it is sufficient for stability that inequalities (Al 1) hold at all points r. REFERENCES W. S. C. & NISBET, R. M. (1975). J. theor. Bid. 52, 441. MACARTHUR, R. H. (1970). Theor. Pop. Biol. 1, 1. MACARTHUR, R. H. (1969). Proc. natn. Acad. Sci. U.S.A. 64, 1369.
GURNEY,
MAY, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton, N.J.: Princeton University Press. RICKLEFS, R. E. (1973). Ecology, London: Nelson. ROUGHGARDEN, J. (1974). Am. Nat. 108, 649. ROUGHGARDEN, J. (1971). Ecology 52,453. &GAL, L. A. & JACKSON, J. L. (1972). J. theor. Biol. 37, 545. WYNNE-EDWARDS, V. C. (1962). Animal Dispersal in Relation to Social Behaviour, Edinburgh and London : Oliver & Boyd.
Spatial Pattern and the Mechanism of Population Regulation W. S. C. GURNEY AND R. M. NISBJS Department of Applied Physics, University of Strathcijde, Glasgow, Scotland (Received 31 July 1975) It has been suggested that the saturation density of many populations is adjusted to match environmental conditions by indirect controls rather than by direct physiological response to lack of some critical resource. This hypothesis is widely regarded as untestable since the observable effects of direct and indirect regulation have been held to be indistinguishable. In this paper we show that the way in which the density of a consumer population competing for a single, spatially heterogeneous, limiting resource, “tracks” variations in its environment is characteristically dependent upon the nature of the density limiting mechanism.
1. Introduction Wynne-Edwards (1962) has contended that the saturation density of animal populations is adjusted to match optimally their environmental conditions by indirect (e.g. behavioural) controls rather than by direct physiological response to lack of a particular resource. It has been widely believed (see, for example, Ricklefs, 1973) that this hypothesis is almost untestable since the observable effects of direct and indirect regulation are essentially indistinguishable. In this paper we show that the way in which the density of a consumer population competing for a single, spatially heterogeneous, limiting resource “tracks” the underlying variations in its environment is characteristically dependent upon the nature of the density limiting mechanism. The problem of the effect of intra-species competition on the way in which population density “tracks” a spatially heterogeneous environment has been considered by Roughgarden (1974) using a “logistic” model analogous to his (1971) model of interspecies competition for a spectrum of resources. His most striking conclusion is that increasing the size of the individual resource capture zone increases the representation of fine scale environmental variations in the pattern of population abundance. This conclusion emerges largely because the Roughgarden model describes only the negative (i.e. 361
362
W.
S.
C.
GURNCY
AND
R.
M.
NISBET
competitive) effects of a finite resource capture zone and neglects the beneficial effects of resource averaging. We have examined the “tracking problem” using a two species model which is a generalization of MacArthur’s (1969, 1970) model of interspecific competition. We consider consumers of density N(r, t) feeding on a single resource of density R(r, t). Since we wish to describe a situation in which a consumer at r feeds on resource in a locality around r, we must be careful about the meaning we ascribe to consumer density. For most plants, and for non-territorial animal species the interpretation is reasonably clear, but in the case of an animal which gathers food over a large but relatively unchanging area we must be careful not to confuse “dispersal motion” with “food gathering motion”. This difficulty, which is discussed in detail by Gurney & Nisbet (1975), may be resolved by regarding N(r, t) as a time average taken over a time short enough to preclude significant dispersal but long enough to average out food gathering trips. We now follow Roughgarden, and describe the geographical extent of the food gathering activities of a consumer at r by means of a normalized resource capture function I(r, r’), whose simplest interpretation is essentially that in a perfectly uniform environment a consumer at r would draw a proportion I(r, r’) dr’ of its total food intake from dr’. We now define the average resource R*(r) available to a consumer at r in a non-uniform environment as R*(r) = .d,, W, WW dr’, (1) and the average feeding load on resources at r as N*(r) = j I(r’, r)N(r’) dr’. allrl
(2)
We assume that resource at position r is replaced at a rate x(&r), R(r)) dependent only upon the local resource density and the underlying favourability pf the locality specified by d(r), and is consumed at a rate F@(r), N*(r)) which depends on the local resource density and the local average feeding load. For simplicity we now define a local net growth rate +Kr), R(r), N*(r)) = x(M), R(r)) -NW, N*(r)), and, making the assumption that the resource species is essentially immobile, write &r> = 7&#@>, R(r), N*(r)). (3) For ease of later interpretation we describe the consumer population at r by a net specific growth rate G(N(r), R*(r)) depending, in some arbitrary way on both the local consumer density N(r) and the average locally available resource R*(r). We envisage the dispersal of individual consumers from r as
SPATIAL
PATTERN
AND
taking place by “jumps” occurring probability P(r, r’) dr’ of landing very simple piece of book-keeping g(r) = G(N(r), R*(r))iV(r)
POPULATION
REGULATION
363
with an average frequency v and having a in dr’. Provided that P(r, r’) = P(r’, r) a now suffices to show that +v J P(r, r’){iV(r’)- N(r)} dr’. (4) all rl
2. Stationary States (A)
UNIFORM
ENVIRONMENTS
In our model the environmental heterogeneity is specified by the spatial variation of the favourability function 4(r). In the case of a homogeneous environment we set 4(r) = &,, and assume that our model has a spatially uniform steady state with R(r) = R, and N(r) = NO such? that G(N,, R,) = 0. d&l, R,, NCJ = 0; (5) In Appendix A, with the aid of the plausible assumption that both the resource capture function I(r, r’) and the jump probability distribution P(r, r’) are functions only of the interval Ir - r’l, we derive completely general conditions for the decay of fluctuations from any such steady state. However, if we consider only systems with “predator-prey” linking between consumer and resource (so that both aG/aR* and -d@N* are strictly positive) and exclude the case in which “autocatalytic” action causes the specific consumer growth rate to increase with consumer density, the stability of any stationary state is assured if, at equilibrium, (6)
and (7) These inequalities are simply a statement of the well known result that any system of interacting populations of the class described by equations (3) and (4) can be stabilized by the provision of sufficient density dependence in either trophic level. It is this result which is the basis of most doubts about the distinguishability of internal and external control. (B)
NON-UNIFORM
ENVIRONMENTS
We now wish to ask what stable population distribution will result from a given pattern of environmental heterogeneity. If the environmental fluctuations are sufficiently small for us to be able to write I * q& for all r, 4(r) = h + Xr> (8) t We note that since Jl(r, r’) dr’ E 1, R(r) = R. +-R*(r) = Rn, and N(r) = NO * N*(r) .= NO.
364
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S.
C.
GURNEY
AND
R.
M.
NISBET
then we can find an approximate analytic solution to this problem, by making the reasonable supposition that if (R,, NO) is the uniform steady state corresponding to 4(r) = &, then R(r) = R,+<(r) and
5 Q R, for all r
(9)
q 4 N, for all r.
W = No + r(r) (10) If, as before, we assume that I(r, r’) and P(r, r’) in fact depend only on the interval Ir-r’l then we see immediately that to a very good approximation the steady state distributions of resources and population are given by the simultaneous solutions of
and
($$W+(f$)
d,, W>Z@ - r’>dr’+ & air,P(r-r%(f) -WI dr’ = 0,
(12)
all partial derivatives being evaluated at (&,, R,, NJ. Equations (11) and (12) are hard to solve directly, but because we have chosen the functions I and P to be functions only of the interval r-r’, we can follow Roughgarden and simplify the process considerably by regarding each function of r as a superposition of sinusoidal components of different “spatial frequency” or wave-vector?. Thus we decompose some function 8(r) into a set of sinusoids O(k) eikar with wave-vector k and amplitude O(k), such that in a d-dimensional universe O(r) = & d j B(k)e ik.‘dk (13) 0 all k The orthogonality of these components enables us to obtain the spectrum e(k) corresponding to a particular O(r) rather easily as O(k) = a/ r 6(r) e-ik.r dr A particularly useful feature of the Fourier transformation equation (14) is that if C(r) = j A(r’)B(r - r’) dr’, all rl then C(k) = A(k)B(k).
(14) defined by (15) (16)
t The wave vector of a plane wave of wavelength 1 is a vector whose direction is normal to the wave front and whose magnitude is 274.l.
SPATIAL
PATTERN
AND
POPULATION
REGULATION
365
With this in mind we multiply both sides of equations (11) and (12) by e -ik.r and integrate over all r, making use of the convolution theorem [equations (15) and (16)] where appropriate. A little routine algebra now tells us that the component of environmental variation with wave-vector k and amplitude X(k) gives rise to a component of population density variation with the same wave-vector and an amplitude ij(k) given by (17) where T(k) =
(18)
($:) (-I$)w +
El-YOGI
The solution of our problem is now formally obtained by plugging (17) back into (13) and performing the requisite integration. However, it is usually more convenient to leave the solution in wave-vector space and deal directly with the “spatial transfer function” T(k). Although expression (18) for this function looks rather daunting we can relatively easily establish some of its more important qualitative properties. To simplify the discussion we shall assume that both the resource capture function Z(r-r’) and the jump probability distribution P(r - r’) are symmetrical Gaussians with -standard deviations (or and g,, respectively. This at once implies that their Fourier transforms are i(k) = exp ( -+o12k2), (19) and P(k) = exp (- +op2k2), (20) immediately enabling us to show that any stable system has a spatial transfer function which goes monotonically to zero as Ikl increases unless an i?G dG ’ _ ii~ + -5;. < “SF* I 0
(21)
-
in which case T(k) goes through a single maximum
before decreasing to zero,
3. Discussion The terms “internal” and “external” control are often somewhat loosely defined, but in the context of our model it is quite clear that external control of the consumer species is represented by @G/JR*), while the degree of internal control is measured by -(aG,@N),.. Since the right hand side of I .a. 24
366
W.
S.
C.
GURNEY
AND
R.
M.
NISBET
inequality (21) must be positive for any non-trivial stable system, we see immediately that this condition for a maximum in the spatial transfer function tells us that the behaviour of a system in which the consumer level is internally regulated (- aG/&V large) is clearly and qualitatively different from the behaviour of a system whose consumer level is regulated externally, provided only that the consumer mobility is not so high as to mask the differences between the two control systems. An illustrative example may serve to illuminate this point. Consider two systems (models A and B) with resource dynamics &r> = R(r)
I - ifi - ajqr)~*(r)a ( > In model A we describe our consumer dynamics by A(r) = [/H*(r)
-YIN(r)
+ v!,P(r
- r’)[N(r’)
(22)
-N(r)]
dr’,
(23)
while in model B the consumers are assumed to behave as N(r) + vs P(r -r’)[N(r’)
- N(r)] dr’.
(24)
Both models are stable for any combination of parameters which produces strictly positive solutions for N(r) and R(r). The difference between them is that in model A control of the consumer level is wholly external (LJG/dN = 0) while in model B the consumers have a measure of internal control. Assuming that the resource capture function Z and the jump probability distribution P are both symmetrical gaussians with standard deviation rrr and gP respectively, equation (18) tells us that in terms of the normalized wave-vector k’ (=cr,k) the spatial transfer function for model A is
while that for model B is
T(k’)
=
--~~-.~
exp ( -IL’)
(26) +
Examples of the form of these two relations as a function of the modulus of normalized wave-vector are displayed in Fig. 1 for various values of the normalized consumer mobility oR[ = u,,/~,]. It is clear from this diagram that
SPATIAL
PATTERN
390
AND
POPULATION
REGULATION
367
-
0
0.6
I.2
Nortnolised
WOYO vector
2-4
3-O
24
30
modulus
(4
0
0.6 Normolised
I.2 wove
I3 vector
modulus
FIG. 1. Typical Spatial Transfer Functions (a) Model with external consumer control only; (b) Model with strong internal consumer control.
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M.
NISBET
if the consumer species is very mobile ((TV > 0,) then the two both have monotonically decreasing transfer functions, but in the low mobility case (ap < CT&the internally regulated consumers show a monotonically decreasing transfer function while those regulated wholly externally show a transfer function with a strong maximum. Lastly we note that only for a model with no consumer internal control and no consumer dispersal does increasing the size of the resource catchment (a& always increase the representation of fine scale environmental differences in the pattern of population density. If the consumers are at all mobile this is true only for wave-vectors below the transfer-function maximum, and if the consumers have either enough mobility or enough internal control to eliminate the maximum from the transfer function then no vestige of this effect remains. 4. C!onclusion We have shown that the relative degree to which environmental variations of differing wavelengths are represented in the equilibrium dispersion of a population supported by a single, spatially heterogeneous, renewable resource is characteristically dependent upon the nature of the mechanism by which the population is stabilized, provided only the population is not so mobile as to mask the relevant effects, and that the variations concerned are not so large as to render a transfer function description unappropriate. An internally controlled or highly mobile population will show a density distribution in which the relative representation of each wavelength of environmental variability decreases monotonically as the wavelength itself decreases. An externally controlled and relatively immobile species will show a density distribution in which some wavelength range is relatively over-represented.
APPENDIX Stability
A
of the Stationary State
In this appendix we analyse the stability of the steady-state solutions of (Al) fib) = +Kr>, R(r), N*(r)>, fi(r) = G(W), R*(r))N(r) +a;i,P(r, r’)(W) - N(r)} dr’. 642) In a homogeneousenvironment, these equations have a uniform steady state with 440, Ro, No) = Wo, R,) = 0. (A3)
SPATIAL
PAT.1.ERN
AND
Let t(r), q(r) be small perturbations
POPULATION
REGULATION
369
in R and N. Then to hrst order 644)
s P(r, r’){N(r’) i =No@+No(g)(*+ ,;,,
- N(r)} dr’,
(A5)
where all partial derivatives are evaluated at the steady state. Provided P(r, r’) depends only on (r--t’J, we can take the spatial Fourier transform of the above equations and obtain W)
Necessary and sufficient conditions for the stability of the system (A6)-(A7) may be derived in the normal manner (see e.g. May 1973, Appendix II) and turn out to be
8G + v(P-1) < 0,
&+“oz and
N h3G-N OaRaN
&iYG
--z OaN* aR*
m2
+vg+)>O.
(A8) W)
These conditions must be satisfied for all wave vectors k if the system (A4)-(A5) is to be unconditionally stable. We now note that for all k, P(k) < 1 and that for “predator-prey” linkage an -
and
aG
i3R’ ’ 0.
With these conditions it is clearly sufficient for stability that in addition a71
- < 0, aR
dG
- < 0, aN
(All)
i.e. the system can be controlled by the provision of sufficient density dependence in either the producer or the consumer dynamics. The inequalities (Al 1) preclude the existence of an inhomogeneous “dissipative” steady state of the type considered by Segal & Jackson (1972). The above proof is not valid for an inhomogeneous environment since the various partial derivatives in equations (A4) and (A5) are then functions of position and we cannot easily Fourier transform the equations. However for
3 70
\V.
S.
C’.
CiURNl.1
ANi)
R.
M.
NISRl:I’
the mildly inhomogeneous steady states analysed in the text, equations (A4)-(A5) are first-order approximations to the perturbation equations at any point r and we conclude that it is sufficient for stability that inequalities (Al 1) hold at all points r. REFERENCES W. S. C. & NISBET, R. M. (1975). J. theor. Bid. 52, 441. MACARTHUR, R. H. (1970). Theor. Pop. Biol. 1, 1. MACARTHUR, R. H. (1969). Proc. natn. Acad. Sci. U.S.A. 64, 1369.
GURNEY,
MAY, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton, N.J.: Princeton University Press. RICKLEFS, R. E. (1973). Ecology, London: Nelson. ROUGHGARDEN, J. (1974). Am. Nat. 108, 649. ROUGHGARDEN, J. (1971). Ecology 52,453. &GAL, L. A. & JACKSON, J. L. (1972). J. theor. Biol. 37, 545. WYNNE-EDWARDS, V. C. (1962). Animal Dispersal in Relation to Social Behaviour, Edinburgh and London : Oliver & Boyd.