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Is the logistic equation a Lotka-Volterra model?
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Ecological Modelling 77 (1995) 95-96
Letter to the Editor
Is the logistic equation a Lotka-Volterra
model?
William Silvert Department of Fisheries and Oceans, Habitat Ecology Diuision, Bedford Institute of Oceanography, P.O. Box 1006, Dartmouth, N.S. B2Y 4A2, Canada
Received 2 August 1993; accepted 6 October 1994
Abstract
In a recent short communication, J.M. Blanc0 (1993) derives equations which he claims refute some widely accepted views about Lotka-Volterra models. These equations are obtained from the logistic equation in the form of a degenerate form of the Lotka-Volterra equation. This note shows that his assertions are unwarranted.
J.M. Blanc0 (1993) has recently claimed to show that the logistic equation is an explicit solution of a two-variable Lotka-Volterra system, from which he deduces some very broad conclusions about Lotka-Volterra models. If correct, these conclusions would invalidate some widely held interpretations of the behaviour of LotkaVolterra models. His analysis proceeds roughly as follows; the logistic equation is
dB/dt=rB(l
-B/K).
(1)
Define a new variable n = (1 - B/K); Blanc0 actually calls this new variable, which he interprets as a resource, N/N,, but for simplicity a single variable can be used. Then Eq. 1 can be rewritten in the form dB/dt
= rBn,
which looks a bit like a Lotka-Volterra equation. To get a second equation we simply differentiate n dn/dt
=d/dt(l = -(r/K)Bn,
-B/K)
= -(l/K)dB/dt
which looks like another Lotka-Volterra equation, and Blanc0 (1993) states that these “two classic models of population dynamics are strongly related” on the basis of this similarity. The question is whether there really is a valid analogy here from which one can derive any general results about Lotka-Volterra equations. Before answering this question, I offer a much simpler way of deriving a pair of equations from the logistic equation that look like Lotka-Volterra equations. Define N = B. This makes N the same variable as B, but with a different name, so we can use N and B interchangeably. We can thus rewrite Eq. 1 in two equivalent ways: dB/dt
= rB - (r/K)BN,
dN/dt
=rN-
(r/K)BN.
This looks even more like the Lotka-Volterra models, since it includes the linear terms. But what does this tell us about the behaviour of Lotka-Volterra models which have two or more independent variables, rather than a single variable to which we give different names? The
0304-3800/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0304-3800(94)00154-5
W. Silvert / Ecological Modelling 77 (1995) 95-96
96
properties of Lotka-Volterra systems have been extensively and rigorously studied (e.g., Goel et al., 19711, and yet Blanc0 claims that the similarity of the equations “refutes the named ‘instability’ or ‘unrealistic behaviour’ of this kind of models”. His paper in no way demonstrates that standard results such as the instability of LotkaVolterra models with an odd number of species (Gael et al., 1971) are invalid. Nor does Blanco’s paper cast any light on the properties of LotkaVolterra trajectories, since the phase space trajectories of his Lotka-Volterra-like equations are simply straight lines, and in Blanco’s version of the equations the system has a fixed point at n = 0, corresponding to the extinction of one species. The value of the Lotka-Volterra equations is that they describe the time evolution of the two or more distinct populations. Before the modern computer era it was customary to apply the term only to certain precise forms satisfying strict requirements (such as an antisymmetric interaction matrix) so that exact analytical results could be produced. Unfortunately, although this led to elegant results, it also excluded the introduction of many realistic considerations, such as the inclusion of self-limiting terms. Contemporary use of computer simulation techniques has rendered this restriction unnecessary, and it is now common to refer to any set of coupled differential equations of the form dxi/dt=fi(.xl...xn),
i=lton,
as “generalized Lotka-Volterra equations”. But, although the mathematics have changed, the equations still describe the evolution of distinct populations. When we start from a single variable, B, and define “new” variables such as n = (1 -B/K) or even N = B, we may end up with equations that look like Lotka-Volterra equations, but because
the system is constrained to lie on just a fixed straight line in phase space, we cannot use this model to draw general conclusions about LotkaVolterra models. In fact, we cannot use these models of a single population to draw any conclusions about the role of interactions in multispecies models. It is difficult to see how one could define a meaningful subset of the models loosely called “Lotka-Volterra” models which correspond to the logistic equation. The fact that two systems are described by equations which look similar does not generally mean that these systems are actually isomorphic to each other, and the equations tell us very little about the dynamics if they do not include all of the constraints involved. In this case the constraints on the system are so strong that they change the dimensionality and totally distort the dynamics in a way that invalidates any inferences based simply on the appearance of the equations. The Lotka-Volterra equations have played a significant role in the development of theoretical ecology, and although there have been many heated debates about whether they are realistic or appropriate for many problems, it is a disservice to the discipline to discount them on such superficial grounds as those presented in Blanco’s paper. Claims that one has refuted decades of research by mathematical manipulation such as simply renaming a variable are bound to provoke confusion and to raise questions about the value of population dynamics models.
References Blanco, J.M., 1993. Relationship between the logistic equation and the Lotka-Volterra models. Ecol. Model., 66: 301-303. Goel, N.S., Maitra, S.C. and Montroll, E.W., 1971. On the Volterra and other nonlinear models of interacting populations. Rev. Model. Phys., 43: 231-276.
Ecological Modelling 77 (1995) 95-96
Letter to the Editor
Is the logistic equation a Lotka-Volterra
model?
William Silvert Department of Fisheries and Oceans, Habitat Ecology Diuision, Bedford Institute of Oceanography, P.O. Box 1006, Dartmouth, N.S. B2Y 4A2, Canada
Received 2 August 1993; accepted 6 October 1994
Abstract
In a recent short communication, J.M. Blanc0 (1993) derives equations which he claims refute some widely accepted views about Lotka-Volterra models. These equations are obtained from the logistic equation in the form of a degenerate form of the Lotka-Volterra equation. This note shows that his assertions are unwarranted.
J.M. Blanc0 (1993) has recently claimed to show that the logistic equation is an explicit solution of a two-variable Lotka-Volterra system, from which he deduces some very broad conclusions about Lotka-Volterra models. If correct, these conclusions would invalidate some widely held interpretations of the behaviour of LotkaVolterra models. His analysis proceeds roughly as follows; the logistic equation is
dB/dt=rB(l
-B/K).
(1)
Define a new variable n = (1 - B/K); Blanc0 actually calls this new variable, which he interprets as a resource, N/N,, but for simplicity a single variable can be used. Then Eq. 1 can be rewritten in the form dB/dt
= rBn,
which looks a bit like a Lotka-Volterra equation. To get a second equation we simply differentiate n dn/dt
=d/dt(l = -(r/K)Bn,
-B/K)
= -(l/K)dB/dt
which looks like another Lotka-Volterra equation, and Blanc0 (1993) states that these “two classic models of population dynamics are strongly related” on the basis of this similarity. The question is whether there really is a valid analogy here from which one can derive any general results about Lotka-Volterra equations. Before answering this question, I offer a much simpler way of deriving a pair of equations from the logistic equation that look like Lotka-Volterra equations. Define N = B. This makes N the same variable as B, but with a different name, so we can use N and B interchangeably. We can thus rewrite Eq. 1 in two equivalent ways: dB/dt
= rB - (r/K)BN,
dN/dt
=rN-
(r/K)BN.
This looks even more like the Lotka-Volterra models, since it includes the linear terms. But what does this tell us about the behaviour of Lotka-Volterra models which have two or more independent variables, rather than a single variable to which we give different names? The
0304-3800/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0304-3800(94)00154-5
W. Silvert / Ecological Modelling 77 (1995) 95-96
96
properties of Lotka-Volterra systems have been extensively and rigorously studied (e.g., Goel et al., 19711, and yet Blanc0 claims that the similarity of the equations “refutes the named ‘instability’ or ‘unrealistic behaviour’ of this kind of models”. His paper in no way demonstrates that standard results such as the instability of LotkaVolterra models with an odd number of species (Gael et al., 1971) are invalid. Nor does Blanco’s paper cast any light on the properties of LotkaVolterra trajectories, since the phase space trajectories of his Lotka-Volterra-like equations are simply straight lines, and in Blanco’s version of the equations the system has a fixed point at n = 0, corresponding to the extinction of one species. The value of the Lotka-Volterra equations is that they describe the time evolution of the two or more distinct populations. Before the modern computer era it was customary to apply the term only to certain precise forms satisfying strict requirements (such as an antisymmetric interaction matrix) so that exact analytical results could be produced. Unfortunately, although this led to elegant results, it also excluded the introduction of many realistic considerations, such as the inclusion of self-limiting terms. Contemporary use of computer simulation techniques has rendered this restriction unnecessary, and it is now common to refer to any set of coupled differential equations of the form dxi/dt=fi(.xl...xn),
i=lton,
as “generalized Lotka-Volterra equations”. But, although the mathematics have changed, the equations still describe the evolution of distinct populations. When we start from a single variable, B, and define “new” variables such as n = (1 -B/K) or even N = B, we may end up with equations that look like Lotka-Volterra equations, but because
the system is constrained to lie on just a fixed straight line in phase space, we cannot use this model to draw general conclusions about LotkaVolterra models. In fact, we cannot use these models of a single population to draw any conclusions about the role of interactions in multispecies models. It is difficult to see how one could define a meaningful subset of the models loosely called “Lotka-Volterra” models which correspond to the logistic equation. The fact that two systems are described by equations which look similar does not generally mean that these systems are actually isomorphic to each other, and the equations tell us very little about the dynamics if they do not include all of the constraints involved. In this case the constraints on the system are so strong that they change the dimensionality and totally distort the dynamics in a way that invalidates any inferences based simply on the appearance of the equations. The Lotka-Volterra equations have played a significant role in the development of theoretical ecology, and although there have been many heated debates about whether they are realistic or appropriate for many problems, it is a disservice to the discipline to discount them on such superficial grounds as those presented in Blanco’s paper. Claims that one has refuted decades of research by mathematical manipulation such as simply renaming a variable are bound to provoke confusion and to raise questions about the value of population dynamics models.
References Blanco, J.M., 1993. Relationship between the logistic equation and the Lotka-Volterra models. Ecol. Model., 66: 301-303. Goel, N.S., Maitra, S.C. and Montroll, E.W., 1971. On the Volterra and other nonlinear models of interacting populations. Rev. Model. Phys., 43: 231-276.