Structural_Sensitivity_ESM - Proceedings of the Royal Society A

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                                       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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