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Electronic Supplementary Material Appendix A Here we prove that there exists an infinite number of functions satisfying (11) such that if and only if conditions (12) are satisfied. To prove that conditions (12) are necessary, we first note that is, by definition, the tangent line to any viable at . Since from (11) any such function must be convex, it must be bounded above by this tangent line. Secondly, we note that is the unique function with second derivative A, and initial conditions determined by the requirements in (11). Since we have specified A as the lowest second derivative of any suitable functional response, it is easy to see – by considering any alternate second derivative – that the admissible functional responses are bounded below by this parabola. Conditions (12) are therefore clearly necessary conditions for to exist. Supp. Fig. 1 illustrates an example of the tangent line and parabola which bound any function satisfying (11). To prove that (12) are sufficient conditions, we need to describe a method that will produce such a function provided that (12) holds. Our method and a resultant function is illustrated in Supp. Fig. 2. For the sake of brevity, in this proof we loosen the constraints on to , but we note that the proof remains valid in the case that the inequality is strict. Firstly, we choose some such that , and (11) holds for and , defined in the same way as and . We first define over . Starting from , we follow the parabola of maximum curvature . At each point along this parabola, we check the tangent line. Initially, this tangent line lies above and it intersects the axis above the origin. As decreases along the parabola, this intersection decreases continuously, so there must come a point at which the tangent line either passes through the origin without crossing first, or must lie tangent to . At this point we let our function follow the tangent line. If it crosses the origin, we are done. If it lies tangent to at some point, we let the function follow . Finally we note that there is a unique point on at which the tangent line passes through 0, so at this point we follow this tangent line to the origin. We note that if we are following , then this point cannot occur before we start: since the derivative of is decreasing, this would imply that at some point the tangent line to the parabola must have crossed the origin. Therefore we are done. For , we can define in a similar manner: we follow the tangent line from and at each point check the parabola of curvature which is tangent to this line. There must be a point at which this parabola is also tangent to (unless the tangent line doesn’t intersect – in which case we are done). At this point we can let our function follow the parabola until it is tangent to , and then follow until it reaches , then we are done. Finally, we note that there are uncountably many valid values of to choose at the start of the method. Therefore an infinite class of viable functions exists Appendix B Here we prove that conditions (19) are necessary and sufficient for an infinite number of functions satisfying (16) to exist, recall: We note that, as in Appendix A, (19) are necessary conditions because is the parabola of minimum second derivative satisfying the initial conditions – so is a lower bound for a suitable function – and is the parabola of maximum second derivative – so is an upper bound. To prove that (19) are sufficient conditions, we need to find a method to construct a valid function which will always work provided (19) are satisfied. We illustrate such a method in Supp. Figs 3 and 4 when absolute and relative distance are used, respectively. For brevity, we will relax the constraints on such that is allowed (as in Appendix A, the proof is still valid for the strict inequality). We consider two cases for : and , where is the base (logistic) function, which has constant derivative. If , then to the left of we initially let the function follow the parabola with (signed maximum) curvature . The parabola of maximum second derivative lies above – so to the left the function either crosses the axis without passing above , in which case we are done, or at some it crosses from below as – in which case we must have . Since the derivative of the parabola is continuous, then by the intermediate value theorem there must be some point where . Whether we’re using definition (3) or (4) for and , the tangent line running parallel to will not intersect either for , so we let the function follow this line and we are done. To the right, if we let the function follow the parabola with (signed minimum) curvature the same holds: this parabola either reaches the or at some it crosses from above as , in which case, again, we must have . If we are using definition (3) for the error, we can use the intermediate value theorem to find a tangent line parallel to , then we can follow this tangent, and we are done. If we are using definition (4), then whether we initially follow the parabola of minimum or maximum curvature to the right depends whether the tangent to the parabolas at intersects the at – in which case we follow the parabola of minimum curvature – or – in which case we follow the parabola of maximum curvature. If we can follow the tangent itself and we are done. In the first (second) case, where this parabola crosses ( ) the tangent line must intersect the at ( ), so since the tangent line varies continuously along the parabola, there must be a point at which the tangent line to the parabola intersects the at . Clearly this tangent line lies between and , so we let the function follow it, and we are done. If , then we can use the same proof if we swap the parabolas the function initially follows to either side (except the parabola initially followed to the right in the case where we are using definition (4)). This is necessary since we require a change in the sign of in order to use the IVT to find a line tangent to . The one difference in this case is that a parabola of negative curvature can intercept the with a positive derivative, but this is easily fixed: since initially, it necessitates a change in the sign of , so we can use the IVT as before Appendix C Here we analytically derive the conditions of when parametric-based analysis of structural sensitivity can provide the same result as the nonparametric test suggested in this paper (we assume the εQ neighbourhood to be small). We consider a function with two parameters and , and aim to find a bound for the domain in space of functions in the neighbourhood of which can be explored by varying the parameters – which is the typical approach taken to structural sensitivity analysis. First we note that taking the Taylor expansion of a function about gives us: and differentiating w.r.t yields: where , and . At we have where and . Since we have and we require , we obtain by rearranging (C3). If we introduce , then by taking linear approximations to , and about in (C2) and discounting second order terms we get: Together, (C4) and (C5) form a linear system of two equations with – provided that we can find expressions for and – two unknowns: and . We can rearrange (C4) to find , and substitute this into (C5) to get: , and . Now, provided that we can find an expression for the bounds of the region in space which corresponds to functions in the neighbourhood of , we can find the analogous region in space. To find such an expression we use condition (3) or (4), depending on which definition of the neighbourhood of we are using. (3) states that for all , so to find the bounds on and , we take the equality here, and using (C1) we obtain: where is the value at which the LHS takes its maximum. If we use condition (4), this becomes , and we need to consider where is again the value at which the LHS takes its maximum. Note, however, that in both cases, may be different for different values of and , so the resulting boundaries may be curvilinear. To avoid this, we choose and use (C6) or (C7) to obtain explicit equations for the two boundary lines in space. This will give us a linear boundary region, and since is a necessary condition for (3) to be satisfied, this linear region must contain the corresponding boundary region for the correct choice of . Appendix D Here we present a method for numerically determining the ‘functional density’ of a point in space for model (9)-(10). We do this by considering all points in the strip between and , and determining whether there is a function with , and such that the curve passes through the point and . The proportion of points between and which do have such functions passing through them then gives us a numerically attainable measure of the volume of functions which project onto the point . Given the values and , and a point , we determine whether there is a function passing through this point as follows: firstly, we check that lies between the minimum and maximum parabolas and used in Appendix B: and . If doesn’t lie between and , there can be no valid . If does lie between these parabolas, then we need to choose a derivative at , , and define the parabolas and , which define the upper and lower bounds for functions passing through with derivative . We need to check several conditions on the 4 parabolas: firstly, we need and (else there are no valid functions having derivative at at all). Secondly, we require that and . These conditions are enough to ensure that there is a function passing through both points that satisfies the restrictions on the second derivative, but in some cases we need one more condition to ensure the existence of such a function which has a negative first derivative for all . If , then in the case that , we need the third condition – in other words, if the minimum of the parabola is at a lower -value than the maximum of the parabola , we require the minimum of to be higher than the maximum of , otherwise any function between and with the given will have to have a positive derivative at some point. If, instead, we have , then the condition is the same with and swapped: if , we require . All of these conditions depend on , and we note that there is always one ‘optimal’ value for each condition to be satisfied – i.e. it is necessary for the condition to hold for this , for it to hold for any derivative at at all – for example, if doesn’t hold when is defined for , it will not hold for any derivative. So we proceed as follows: starting from , we check all the conditions. If one of the conditions is not satisfied, then we either decrease or increase by a small increment until either the condition is satisfied or the optimal is reached without satisfying the condition – in which case we can conclude that there is no valid function passing through . If the given condition is satisfied, we continue checking the others and making adjustments until either we find a such that all conditions are satisfied – in which case we are done – or we find an unsatisfied condition which, in order to rectify, we have to undo the changes we have already made to to ensure it satisfies another condition – in which case there can be no valid function. If at any point we reach either the optimal for a given condition (or a sufficiently large/small when the optimal derivative is ) without satisfying it, we should stop. Appendix E - Age-structured Predator-Prey Model in a chemostat with nutrient Here we provide an addition demonstration of our method by demonstrating structural sensitivity in a predator-prey model in a chemostat with nutrient. We consider the following model (Fussmann et al., 2000): where N is the concentration of nutrient in the chemostat, C is the concentration of green algae, R is the concentration of reproducing plankontic rotifer and B is the total rotifer concentration. We consider all parameters to be the same as in (Fussmann et al., 2000), except for , which we change slightly from 2.25 to 1.95, and , for which we consider several values. We consider the functional response of the green algae utilising the nutrient, , to be unspecified, but require that it is a typical Holling type II function, and so satisfies conditions (11) in Section 3.1 of the main text. We will investigate the structural sensitivity of this system in terms of variation of the stability of the nontrivial equilibrium due to changes in , taking the base functional response to be the one used in the text: , where and . We note that since the nontrivial equilibrium concentration is given by E3, equation E1 determines the relationship between and , so if, for example, we take as a parameter, our choice of will determine the value , and the other equilibrium values and will follow. Determining the stability of depends on the Jacobian, so we need to specify the parameter to complete our analysis. To determine the -neighbourhood of the base function in ( - ) parameter space, we can adapt conditions (12) to give us the necessary and sufficient conditions for the existence of a function satisfying (11) such that its derivative at is equal to : , , (E5) and . Where and . Therefore our analysis simply consists of scanning ( - ) space and checking these conditions to obtain the - neighbourhood, and then checking the sign of the eigenvalues of the corresponding Jacobian to determine the stability of . Supplementary Figure 5. shows the results of such an investigation for three values of the chemostat dilution rate, : (A) , (B) and (C) . The green, red, azure and dark blue regions have the same meanings as in Figs 3 and 5 in the main text. From this Figure, we see that the model exhibits structural sensitivity in all three cases to some extent, although it is more pronounced for intermediate values of . For the parameters and choice of used by Fussmann et al. (2000), this model exhibits forward and backward Hopf bifurcations when is varied, which is shown by the decrease and subsequent increase in the stability region as is changed from 0.175 to 0.5 to 0.7. However, since in all three cases there are significant regions of both stability and instability present, we can conclude that for certain choices of , these bifurcations simply would not be observed. However, the fact that the azure region crosses the bifurcation line in all cases indicates that in this case, this fact could be demonstrated by fixing the parameterization and varying the parameters and . Supplementary Figure Captions 1. The tangent line , and parabola which form the upper and lower bounds of a function taking the values , and satisfying conditions (11). There will exist at least one such function in the -neighbourhood of (i.e which stays between and ) if and only if the tangent lies above and the parabola lies below over the whole interval . 2. Example of a functional response satisfying criterion (11) constructed using the method described in Appendix A – depending only on conditions (12) being satisfied. The light dashed curves represent the tangent line at , and the parabola with second derivative that is tangent to the function at . These form upper and lower bounds, respectively, on any potential function. See the text for more details of the method. 3. Example of a growth rate satisfying criterion (16) constructed using the method described in Appendix B for the case in which definition (3) of closeness (absolute closeness) is used. The method depends only on conditions (19) being satisfied. The thin solid parabolas are those of maximum and minimum (signed) curvature, which are tangent to the growth function at . These form upper and lower bounds, respectively, on any potential function. See the text for more details of the method. 4. Example of a growth rate satisfying criterion (16) constructed using the method described in Appendix B for the case in which definition (4) of closeness (relative closeness) is used. The method depends only on conditions (19) being satisfied. As in Supp. Fig. 2, the thin solid parabolas are those of maximum and minimum (signed) curvature, which are tangent to the growth function at . These form upper and lower bounds, respectively, on any potential function. See the text for more details of the method. 5. Testing structural sensitivity of chemostat model (E1)-(E4) to variation of the functional response of the green algae. The whole space of functions is projected onto the ( - ) space. The εQ neighbourhood of h is defined based on the relative difference between h and its perturbations. Three values of the chemostat dilution rate are used: (A) (B) and (C) , for all other parameters see the text. Red and green domains describe, respectively, the unstable and stable stationary state. The blue domain corresponds to the region covered by conventional sensitivity analysis obtained by varying only parameters and in the base function.

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posted: | 4/27/2013 |

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