On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
On-line Supplementary Material
Overview:
Appendix A1: Basic results from the Consumer-Resource Lotka-Volterra model
Includes reference to Fig. 6
Appendix A2: The role of attack rate and new IS for the stability response of the
Rosenzweig-MacArthur Consumer-Resource model when eigenvalues are
complex
Appendix A3: Note on how energy flux is affected by carrying capacity and conversion
efficiency in the Consumer-Resource models
Appendix A4: Note on how attack rates affect the Consumer/Resource biomass ratio
Appendix A5: Changing intrinsic resource growth in the Consumer-Resource models
Includes reference to Fig. 7
Appendix A6: Changing consumer mortality in a Consumer-Resource system
Includes reference to Fig. 8
Appendix A7: Changing consumer mortality in a food chain
Includes reference to Fig. 9
Appendix A8: The relationship between total flux and stability when changing consumer
attack rate in a food chain
Includes reference to Fig. 10
Appendix A9: Lotka-Volterra food chain with the predator present
Includes reference to Fig. 11
Appendix A1: Basic results from the Consumer-Resource Lotka-Volterra model
The Lotka-Volterra model (Fig. 6) showed very similar results to the RM model (Fig. 2c-d) in
terms of per capita and total fluxes when increasing acr.
Appendix A2: The role of attack rate and new IS for the stability response of the
Rosenzweig-MacArthur Consumer-Resource model when eigenvalues are complex
Here we consider the behavior of the λmax expression for the Rosenzweig-MacArthur model
under the condition that eigenvalues are complex. We consider two cases for the complex
eigenvalues. First, when λmax is a complex number and the real part is negative (i.e. the
dynamics have a locally stable equilibrium) and, second, we will briefly discuss when λmax is a
1
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
complex number and the real part is positive. This latter case is less interesting as the solution
is no longer on the equilibrium but rather becomes a limit cycle so the dynamic behavior is no
longer well explained by the eigenvalues.
For starters, let us assume that the dominant eigenvalue for the Rosenzweig-MacArthur
Consumer-Resource is complex and therefore look at the local stability property of the real
part of the solution alone (λmax when real; λMR):
λMR=
−𝑎𝑐𝑟 𝑒𝑏𝑚𝑐 𝑟+𝑎𝑐𝑟 𝑒𝐾𝑚𝑐 𝑟−𝑏𝑚𝑐 2 𝑟−𝐾𝑚𝑐 2 𝑟
(A2.1a)
2𝑎𝑐𝑟 𝑒𝐾𝑎𝑐𝑟 𝑒−𝑚𝑐 )𝑚𝑐
The full representation of the dominant eigenvalue can be achieved by combining eq. A2.1 b
with “what turns complex”, i.e. eq. A2.1c (here a = acr).
λmax =
−𝑎𝑐𝑟 𝑒𝑏𝑚𝑐 𝑟+𝑎𝑐𝑟 𝑒𝐾𝑚𝑐 𝑟−𝑏𝑚𝑐 2 𝑟−𝐾𝑚𝑐 2 𝑟 (+/−)"𝑡𝑢𝑟𝑛𝑠 𝑐𝑜𝑚𝑝𝑙𝑒𝑥"
2𝑎𝑐𝑟 𝑒𝐾(𝑎𝑐𝑟 𝑒−𝑚𝑐 )
(A2.1b)
√(𝑎𝑒𝑏𝑚𝑟 − 𝑎𝑒𝐾𝑚𝑟 + ℎ𝑚2 𝑟 + 𝐾𝑚2 𝑟)2 − 4(𝑎4 𝑒 4 𝐾 2 𝑚𝑟 − 𝑎3 𝑒 3 𝑏𝐾𝑚2 𝑟 − 3𝑎3 𝑒 3 𝐾 2 𝑚2 𝑟 + 2𝑎2 𝑒 2 𝑏𝐾𝑚3 𝑟 + 3𝑎2 𝑒 2 𝐾 2 𝑚3 𝑟 − 𝑎𝑒𝑏𝐾𝑚4 𝑟 − 𝑎𝑒𝐾 2 𝑚4 𝑟)
(A2.1c)
For what we want to show - that the “complex domain” responds to attack rate and more
generally, new IS, in a manner consistent with the Lotka-Volterra result, we do not need to
consider the complex part (i.e., this part is the square root term in the quadratic solution, eq.
A2.1c). As done for the Lotka-Volterra system in the main text, this separation allows us to
consider the strength of the real part of the eigenvalue (A2.1a) when the system is complex.
Note, the A.2.1a term entirely determines the stability strength when the eigenvalue is
complex.
The main result of this proof is straightforward: that is, that when we have a negative
eigenvalue, increasing a (later we show this is true for new IS in general) always destabilizes
the system when it is in the complex phase and the equilibrium is stable (A.2.1a). This follows
because when we have a complex negative eigenvalue the real part can be can be described by
rearranging eq. A2.1 to the following:
λMR =
𝑚𝑐 𝑟(−𝑎𝑐𝑟 𝑒𝑏+((𝒂𝒄𝒓 𝒆−𝒎𝒄 )𝑲−𝒃𝒎𝒄 ))
(A2.2)
2𝑎𝑐𝑟 𝑒𝐾(𝑎𝑐𝑟 𝑒−𝑚𝑐 )
2
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
λMR =
𝑚𝑐 𝑟(−𝑏+(𝑲−
𝒃𝑚𝑐 𝑲𝑚𝑐
−
))
𝑎𝑐𝑟 𝒆 𝑎𝑐𝑟 𝒆
(A.2.3)
2𝐾(𝑎𝑐𝑟 𝑒−𝑚𝑐 )
From A.2.3 we can see that if the real part is negative, then given that K>R* then by
substitution K>bmc/(acre-mc) and so K> bmc/(acre)+K mc/(acre) in A.1.3. Thus, increasing
"acre" makes K dominate more in A.2.3 (bold part in A.2.3) and therefore, since this term is
positive, it necessarily reduces the negativity of the eigenvalue (i.e. it becomes less stable).
Since “ae” also simultaneously reduces the denominator then we know that increasing "ae"
necessarily destabilizes the system when the eigenvalue is complex with a negative part. This
result is entirely consistent with the Type I (LV) result shown in the paper.
If we consider the new IS for the Type II (RM) case, after substitution of R* and rearranging,
we find:
𝑁𝑒𝑤 𝐼𝑆 =
𝐾(𝑎𝑐𝑟 𝑒−𝑚𝑐 )
(A2.4)
𝑚𝑐 𝑏
and so we can rearrange eq. A.2.3 to show:
𝑟
λMR = − 2 𝑁𝑒𝑤 𝐼𝑆 +
𝒃𝑚𝑐 𝑲𝑚𝑐
−
))
𝑐𝑟 𝒆 𝑎𝑐𝑟 𝒆
𝑚𝑐 𝑟(𝑲−𝑎
(A.2.5)
2𝐾(𝑎𝑐𝑟 𝑒−𝑚𝑐 )
Thus, similar to the Lotka-Volterra case, new IS is correlated with λMR for the entire negative
complex phase of the eigenvalues such that increasing new IS drives destabilization. Thus, we
can say that when the eigenvalue is complex and stable, increasingly strong control (new IS)
correlates with destabilization. Again, this result is entirely consistent with the Type I LotkaVolterra case, which only produces complex, but negative (stable), eigenvalues.
The positive eigenvalue is not as clear nor as informative. As discussed above, once positive,
the eigenvalue has little to do with the non-equilibrium attractor. However, in this case, the
eigenvalue can get more positive with increasing acr or new IS, depending on whether the
numerator or denominator dominates. Clearly, at extremely high attack rate or New IS, the
numerator approaches K and the denominator continues to grow, therefore dominating the
solution. At this point, increases in attack rate, or New IS, actually make the eigenvalue less
positive.
3
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
Appendix A3: Note on how energy flux is affected by carrying capacity and conversion
efficiency in the Consumer-Resource models
The effects of increasing the carrying capacity (K) are straight forward, total energy flux
increases with increasing K. We use the partial derivative of the equations describing total
energy flux in Table 1 to demonstrate this. For the Lotka-Volterra case, the partial derivative
with respect to K is positive and increasing K will always have a positive effect on the total
energy flux (eq. A.3.1). The partial derivative for the Rosenzweig-MacArthur model will also
be positive when ae>m, which is a prerequisite for a existence of a consumer, and increasing
K will hence have the same positive effect on total energy flux.
𝑚𝑐 𝑟
(A.3.1)
𝑎𝑐𝑟 𝑒𝐾
𝑏𝑚𝑐 2 𝑟
(A.3.2)
𝐾2 (𝑎𝑐𝑟 𝑒−𝑚𝑐 )2
Increasing New IS by increasing the conversion efficiency (e) affects the total energy flux in
the same way as for when decreasing mortality, more energy will be accumulated in the
predator biomass (not shown, McCann 2012).
Appendix A4: Note on how attack rates affect the Consumer/Resource biomass ratio
Here we look at the response in Consumer/Resource biomass ratio to changes in consumer
attack rate. We use the partial derivatives of the expressions for the C/R ratios from Table 1 in
the main paper. For the Lotka-Volterra model increasing attack rate (acr) will always increase
the C/R ratio (eq. A4.1). This can be seen by the positive derivative. For the RosenzweigMacArthur model we get the same result, but this is dependent on that acre>mc (eq.A.4.2).
However, this condition is also required for the consumer to have a positive equilibrium, so
we can conclude that the C/R ratio increases with increasing acr for the realistic parameter
range.
𝑟
(A4.1)
𝑎𝑐𝑟 2 𝐾
𝑏𝑒 2 𝑟
(A4.2)
𝐾(−𝑎𝑐𝑟 𝑒+𝑚𝑐 )2
Appendix A5: Changing intrinsic resource growth in the Consumer-Resource models
New IS does not incorporate the intrinsic resource growth (r), however, we have shown that r
is also important for stability (real part eq. 7, 8 main text). For the Lotka-Volterra case it is
easy to see that r works in the opposite direction from aeK when λmax is complex, i.e.
4
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
increasing r acts stabilizing (eq. 7; -mcr/2acreK, Fig. 7 a, full lines). Interestingly, in this
scenario r and K, that both can be considered productivity measures, have opposite effects.
While less obvious from the equations, when λmax is real (and the real part is described by the
full eq. 7) increasing r has a destabilizing effect (Fig. A7 a, dotted lines). The effect of r
changes as soon as λmax goes from complex to real, seen in the checkmark shapes in Fig. 7 a.
The Lotka-Volterra model with logistic resource growth that we use is always stable.
Correspondingly, in the Rosenzweig-MacArthur case, when the system is stable and λmax is
real, then increasing r acts destabilizing (Fig. 7 b, second half of full line). While when the
system is stable and we have a complex λmax, increasing r has a stabilizing effect (Fig. A5.1 b,
dash-dotted line). When the system is unstable (and λmax is complex) r has a destabilizing
effect (Fig. 7 b, dashed line).
It is worth noting that the C/R ratio increases in all cases presented irrespectively of the
stability response an increasing r always results in increasing consumer density while the
resource stays constant (not shown).
Appendix A6: Changing consumer mortality in the Consumer-Resource models
When we increase consumer mortality (mc) it only affects the position of the consumer
isocline, not the resource isocline (see eq. 5.1-5.2, 6.1-6.2). The responses in biomasses from
changing mc can be found in Fig. 8 (a-b). For both models, the total energy flux between
consumer and resource shows a hump-shaped relationship to mc (Fig. 8 c-d). In the LotkaVolterra model this is not translated into a hump shape in consumer density (Fig. 8 a), due to
that the increase in mortality more than compensates for the increased production, and the
consumer density only decreases. While in the Rosenzweig-MacArthur case we do get a
hump shaped pattern in consumer density (Fig. 7 b). This means that a hydra effect is present,
i.e. increasing consumer mortality can result in increasing consumer densities (Abrams 2009).
We argue this is due to that the consumer isocline intersects the resource isocline closer to the
maximum production of the resource, as for our paradox of attack rate example in the main
paper.
Concerning stability; when increasing acr (main paper Fig. 2-3), more energy generally results
in a less stable system as soon as the system enters the complex phase. This is followed by a
phase where there is a decrease in total flux, but the system is still turning less stable (Fig.
2b). If we now compare this to the case when we are decreasing IS by increasing mc; total
energy flux may increase due to the hydra effect, but the system only gets more stable during
5
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
this phase (Fig. 8). One interesting aspect is hence that we can have both scenarios; less
energy does not necessarily lead to a more stable system (as for increasing acr in Fig. 2) and
more energy may not necessarily lead to a less stable system (as for increasing mc in Fig. 8).
Appendix A7: Changing consumer mortality in a food chain
As mentioned, changing consumer mortality only affects the consumer isocline for both
models. When the predator is present, changing mortality (mc) levels will only result in that
the equilibrium point will move along the predator density axis, as the intersection between
the consumer and predator isoclines will not change. Since the equilibrium densities of
consumer and resource are fixed (Fig. 9), the relationship between the equilibrium density of
the resource (R*) and K is not going to change. From this follows that the equilibrium point
cannot move in relation to the maximum production of the resource (K/2), and we cannot get
a hydra effect by changing consumer mortality when a predator is present. This is relevant for
the Rosenzweig- MacArthur case. Note, however, that the hydra effect is important for the
invasion of the predator in the Rosenzweig-MacArthur model as we show in the main paper
(Fig. 4).
When equilibrium densities of the consumers are locked, this means that the energy flux
between resource and consumer will also be locked (Fig. 9). Only the relative fluxes away
from the consumer, to background mortality (-mcC) and mortality imposed by the predator (apcCP), changes.
Increasing consumer mortality results in less energy transfer between
consumer and predator, decreasing predator densities, and eventually the extinction of the
predator. Still, the flux to the predator affects stability in the P-C link as expected, i.e. it gets
more stable when consumer mortality increases and the flux to the predator decreases (Fig. 9).
The main message here is that the hydra effect disappears when we have a predator present.
We can, however, get a hydra effect when imposing mortality on the predator (if it has a type
II functional response, not shown). It is also quite likely that the hydra effect (arising from
changing mc) can reemerge if we add a 4th trophic level.
Appendix A8: Total flux and stability patterns when changing consumer attack rate in a
food chain
When increasing acr in a food chain the stability measure shows a typical checkmark pattern with a
transition from an un-excitable to an excitable phase (Fig. 5 a-b main text) and the mismatch between
stability and total energy flux at high attack rate remains (Fig. 10). Again, there is a correspondence
between total flux and stability during the non-excitable phase (dotted lines Fig. 10), and during the
6
On-line Supplementary Material: Nilsson and McCann: Interaction strength revisited
initial part of the excitable phase (positive relationship, full lines). Then as attack rate (acr) is increased
the relationship switches, resulting in a hump shaped relationship between total energy flux and local
stability (Fig. 10). We show the consumer-resource flux in Fig. 10, but as described in the main text, it
corresponds completely to the predator-consumer flux (the consumer density is constant).
Appendix A9: Lotka-Volterra food chain with the predator present
The Lotka-Volterra model (Fig. 11) showed very similar results to the RM model (presented
in the main paper Fig. 5 a-b) in terms of how per capita and total fluxes change when
increasing acr in a food chain scenario. However, the mismatch between total flux and
stability is more pronounced in the LV case (Fig. 11).
7