The role of convection in the existence of wavefronts for biased movements

We investigate a model, inspired by (Johnston et al., Sci. Rep., 7:42134, 2017), to describe the movement of a biological population which consists of isolated and grouped organisms. We introduce biases in the movements and then obtain a scalar reaction-diffusion equation which includes a convective term as a consequence of the biases. We focus on the case the diffusivity makes the parabolic equation of forward-backward-forward type and the reaction term models a strong Allee effect, with the Allee parameter lying between the two internal zeros of the diffusion. In such a case, the unbiased equation (i.e., without convection) possesses no smooth traveling-wave solutions; on the contrary, in the presence of convection, we show that traveling-wave solutions do exist for some significant choices of the parameters. We also study the sign of their speeds, which provides information on the long term behavior of the population, namely, its survival or extinction.


Introduction
In this paper we investigate a model to describe the movement of biological organisms.Its detailed presentation appears in Section 2. Inspired by the recent paper [8], we assume that the population is constituted of isolated and grouped organisms; our discussion is presented in the case of a single spatial dimension but could be extended to the whole space.The first rigorous mathematical deduction of movement for organisms appeared in [17]; since then, several models have been proposed, see for instance [7,8,13,14,15] and references there.In this context, a common procedure is to start from a discrete framework where the transition probabilities per unit time τ and for a one-step jump-width l are assigned, and then pass to the limit for τ, l → 0. In the aforementioned papers the limiting assumptions make the diffusivity totally responsible for the movement, and no convection term appears; see however [14, §5.3] and [16], for instance, for the deduction of a model which also include a convective effect.Here, we generalize the model in [8] by introducing a possibly biased movement, which leads, in general, to a convective term.As a consequence, we show the appearance of a greater variety of dynamics which allow to better investigate the long term behavior of the population; in particular, to predict its survival or extinction.
Our model is described by a reaction-diffusion-convection equation where the functions f, D and g satisfy (2.10), (2.11) and (2.12), respectively.The unknown function u denotes the density (or concentration) of the population and then it has bounded range; for simplicity we assume u ∈ [0, 1].An interesting feature of equation (1.1) in this context is that negative diffusivities arise for several natural choices of the parameters.As in [8], here we consider a diffusion term which makes equation (1.1) of forward-backwardforward type.This occurrence was already noticed in other papers, see for instance [18,20] in the case of a homogeneous population under different assumptions.Notice however that the deduction of the model both in [8] and in the present paper also involves the reaction term, while in [18,20] it is limited to diffusion.As opposite to positive diffusivities, which model the spatial spreading, negative diffusivities are usually interpreted to model the "chaotic" movement which follows from aggregation [18,20].In turn, the latter is "a macroscopic effect of the isolated and the grouped motility of the agents, together with competition for space" [11].At last, we assume that the reaction term g shows the strong Allee effect, i.e., it is of the so called bistable type (see assumption (g) below).We focus on the existence of traveling-wave solutions u(x, t) = ϕ(x − ct) to equation (1.1), for some profiles ϕ = ϕ(ξ) and wave speeds c, see [6] for general information.If the profile is defined in R, it is monotone, nonconstant, and reaches asymptotically the equilibria of (1.1), then the corresponding traveling-wave solution is called a wavefront.We consider precisely decreasing profiles which connect the outer equilibria of g, i.e., ϕ(−∞) = 1 and ϕ(∞) = 0. (1. 2) The case when profiles are increasing, and then satisfy ϕ(−∞) = 0, ϕ(∞) = 1, is dealt analogously and leads to a similar discussion.These solutions, even if of a special kind, have several advantages: they are global, they are often in good agreement with experimental data [13], and can be attractors for more general solutions [5].Moreover, when u represents the density of a biological species, as in this case, then condition (1.2) means that, for times t → ∞, the species either successfully persists if c > 0, or it becomes extinct if c < 0. The wavefront profile ϕ must satisfy the ordinary differential equation We used the notation ˙:= d/du and ′ := d/dξ.Although one can consider the case of discontinuous profiles, see [10,11] and references there, in this paper we focus on regular monotone profiles of equation (1.3).This means that they are continuous, and of class C 2 except possibly at points where D vanishes; then solutions to equation (1.3) are intended in the distribution sense.
The existence of wavefronts is treated here in a quite general framework, which includes, in particular, our biological model.More precisely, we fix three real numbers α, β, γ satisfying and assume, see Figure 1, , and D < 0 in (α, β); Since f in (1.1) is defined up to an additive constant, we can take f (0) = 0.The term ḟ (u) represents the drift of the total concentration u and prescribes in particular if a concentration wave is moving toward the right ( ḟ (u) > 0) or toward the left ( ḟ (u) < 0).The parabolic equation (1.1) is of backward type in the interval (α, β) and of forward type elsewhere; moreover, it degenerates at α and β.
The presence of wavefronts to (1.1) satisfying (D) and (g) and with f = 0 was first discussed in [9], where it is shown in particular that, if a wavefront exists, then γ / ∈ [α, β].Such a situation and many others, again with f = 0, was also considered in [8, cases 6.3, 8.3], in the framework of the particular model deduced in that paper.The case with convection is not yet completely understood.Then our issue here is whether and when the presence of the convective flow allows the existence of wavefronts.
An intuitive argument, see Remark 3.3, shows that the answer is in the affirmative at least for suitable concave f .We now briefly report on the content of this paper.
In Section 2 we introduce the biological model and state our main results about it, for more immediacy, in a somewhat simplified way; proofs, which require the analysis of the general case dealt in Section 3, and more details, are deferred to Section 4. In particular, we provide positive or negative results for each of the possible behaviors of f , namely, when f is concave, convex, or it changes concavity once.
In Section 3 we investigate the fine properties (uniqueness, strict monotonicity, estimates of speed thresholds) of such wavefronts for equation (1.1) satisfying (f)-(D) and (g).A similar discussion for a monostable reaction term g appeared in [1] and [3], respectively in the general framework and for the population model with biased movements.We recall that g is called monostable if g > 0 in (0, 1) and g(0) = g(1) = 0.
As in our aforementioned papers, we exploit here an order-reduction technique.Since we focus on profiles ϕ = ϕ(ξ) that are strictly monotone when ϕ ∈ (0, 1), we can consider the inverse function ϕ −1 (ϕ) of ϕ and, by denoting z(ϕ) := D(ϕ)ϕ ′ ϕ −1 (ϕ) , we reduce the problem (1.3) to a first-order singular boundary-value problem for z in [0, 1].This problem is tackled by the classical techniques of upper-and lower-solutions.This technique requires lighter assumptions than the phase-plane analysis in [8] and is simpler than the geometric singular perturbation theory exploited in [10].Then wavefronts satisfying (1.2) are obtained by suitably pasting traveling waves.The results appear in Section 3, they are given for an arbitrary equation (1.1) satisfying conditions (f), (D), (g), and they are original.About (g), the mere requirement that g is continuous and the product Dg differentiable at 0 would be sufficient for us.Both for (D) and (g), we made slightly stricter assumptions than necessary both for simplicity and because they are satisfied by our biological model with biased movements.The cases when the internal zero of g is before α, i.e. γ ∈ (0, α), or after β, that is γ ∈ (β, 1), are not treated here.Equation (1.1) with f = 0 admits wavefronts in these cases and we expect that they persist also in the presence of the convective effect f .The issue of the linear stability of the wavefronts is certainly interesting; we claim that it could be developed as in [10,19], with a similar discussion.

A biological model with biased movements
In this section we first summarize a model for the movement of organisms recently presented in [8] for populations constituted by two groups of individuals.Then we show how a convective term can appear in the equation because of a biased movement.At last, we provide our results about wavefronts for such a model; proofs are deferred to Section 4.

The model
The population is divided into isolated and grouped organisms.Both groups can move, reproduce and die, with possibly different rates.The organisms occupy the sites jl, for j = 0, ±1, ±2, . . .and l > 0; we denote by c j the probability of occupancy of the j-th site.Let P i m and P g m be the movement transitional probabilities for isolated and grouped individuals, respectively; we use the notation P i,g m to indicate the two sets of parameters together.Analogously, the corresponding probabilities for birth and death are P i,g b and P i,g d .Differently from [8], we also introduce the parameters a i , b i ≥ 0 and a g , b g ≥ 0, which characterize a (linearly) biased movement for the isolated and grouped individuals.For the isolated individuals the bias is towards the left if a i − b i > 0 and towards the right if a i −b i < 0; for the grouped individuals the same occurs when either a g −b g > 0 or a g −b g > 0, respectively.In the case of [8] one has a i = b i = a g = b g = 1 and then a i,g − b i,g = 0.Then, the variation δc j of c j during a time-step τ > 0 is given by where By noticing that every bracket is divided by 2, we deduce since in the deduction of (2.5) a bias a i,g implies a converse bias b i,g = 2 − a i,g .The continuum model is obtained by replacing c j with a smooth function c = c(x, t) and expanding c around x = jl at second order.Then, we divide (2.5) by τ and pass to the limit for l, τ → 0 while keeping l 2 /τ constant; for simplicity we assume l 2 /τ = 1.To perform this step one makes the following assumptions on the reactive-diffusive terms [8]: The above limits define the diffusivity parameters D i,g , the birth rates λ i,g , and the death rates k i,g ; all these parameters are non-negative.About the convection terms, we require for some C i,g ∈ R. We stress that the parameters C i,g can be either positive or negative according to the values of the bias coefficients a i,g and b i,g ; in particular, if C i > 0 then we have a bias toward the left of the isolated individuals, and toward the right if C i < 0; the analogous bias for the grouped individuals corresponds to either C g > 0 (left) or C g < 0 (right).If C i,g = 0, then the corresponding bias is too weak to pass to equation (2.9); with a slight abuse of terminology we say that the corresponding group has no convective movement.At last, assumption (2.8) 1 is compatible with (2.6); assumption (2.8) 2 is analogous to (2.7) 4 .
In conclusion, we obtain the equation with (2.12) The model (2.9) depends on the eight parameters C i,g , D i,g , k i,g and λ i,g .Equation (2.9) coincides with (1.1) but we agree that when we refer to (2.9) we understand f , D and g as in (2.10)-(2.12).We point out that f (0) = f (1) = 0, i.e., the convective flow vanishes when the density is either zero or maximum, as physically it should be.When C i = 0 the isolated individuals have no convective movement and the function f is convex in [0, 1] if C g > 0 and concave otherwise.Instead, when C g = 0 the grouped individuals have no convective movements and f changes its concavity for u = 2/3.The diffusion and reaction terms (2.11)-(2.12)coincide with those in [8, (2)], while f is missing there.

Main results on the model
About the model introduced in the previous subsection, the case we are interested in is when conditions (D) and (g) are satisfied; the corresponding assumptions on the parameters have already been given in [8,11].
In this case we have

.13)
The reaction term g in (2.12) satisfies (g) if and only if k g = 0, λ g > 0 and ) Notice that ω ∈ (0, 1), β − α = 2ω/3 and α + β = 4/3.The condition k g = 0 clearly has no biological sense, and is interpreted in the sense that the life expectancy of grouped individuals is much larger than that of isolated individuals; the condition r i > 0 further hinders the latter.Here, γ is the Allee parameter [8,11].
Here follows a simple necessary condition for the existence of wavefronts.
We now summarize the restrictions required on the parameters: with ω defined in (2.13).We always assume conditions (2.15)-(2.17) in the following, without any further mention.The results below are preferably stated by referring to the following dimensionless quotients and by lumping the parameters referring to the grouped population into a single dimensionless parameter as follows: Under this notation we have Notice that E g gathers the parameters concerning convection, diffusion and reaction of the grouped individuals; the parameter µ is the ratio between the net increasing rate of the isolated and grouped individuals.Notice that condition (2.17) is equivalent to

.20)
The convective term f can change convexity at most once; then, it can be either concave or convex, or else convex-concave or concave-convex.We now examine each of these cases; in all of them, we emphasize that s is always multiplied by d.Since the parameter s does not depend on d, we can understand sd as a variable independent from d, which lumps the ratios of the coefficients related to the movement.In this way we shall often deal with the couple (ω, sd) of parameters, where ω depends on d.

The concave case
The convective term f is strictly concave if and only if (see Lemma 4.2).In this case, model (2.9) admits wavefronts satisfying (1.2) and we can also discuss the sign of their speeds c; we now provide some prototype results.The key condition is (2.22) Condition (2.22) contains several parameters; therefore, there are many ways of discussing the results, depending on which parameters are set and which are held constant; we focus on two different choices.
First, for fixed C g , D g we consider the triangle, see Figure 3 on the left,  Moreover, for such (r i , λ g ) ∈ T g (d) we have, see Figure 4: where τ is defined in (4.15), then (2.9) admits wavefronts satisfying (1.2) with positive speeds (see Remark 4.6); provides profiles and all of them have negative speeds (see Remark 4.7).Second, notice that, assuming again (2.21), we can interpret (2.17) and (2.22) as relationships between d and µ for fixed E g , see Figure 3 on the right and Corollary 4.1.In this framework, profiles exist for every couple (d, µ) lying in the region between the red and blue lines, and below the black line.
The convex case If f is convex, then equation (2.9) admits no such wavefronts.Indeed, we show a stronger result: wavefronts satisfying (1.2) never exist if f is convex just in the interval [α, β] (see Remark 4.3).
The case when f changes convexity We now consider the two cases when f changes convexity; in order to simplify the analysis, we focus on the particular case when the inflection point of f coincides with γ.Recall that γ represents the Allee parameter [8], which describes the threshold separating a decrease of concentration (if u < γ) from an increase of concentration (if u > γ).The assumption that f has an inflection point at γ means that the maximum drift ḟ (if f is convex-concave) or the minimum drift (if f is concave-convex) is precisely reached at γ.We refer to Figure 5 for an illustration of both cases.• Assume f is convex-concave and define the set (2.25) If (ω, sd) ∈ S and estimate (4.27) holds, then equation (2.9) has wavefronts satisfying condition (1.2).
• Assume f is concave-convex and define the set

Theoretical results
In this section we provide the theoretical results that are needed for the investigation of model (2.9).In the following we consider equation (1.1) and we always assume (1.4) and (f), (D), (g), without any further mention.The existence of a wavefront solution to (1.1), whose profile satisfies (1.2), is obtained by pasting profiles connecting 0 with α, α with γ, γ with β, and β with 1.Each subprofile exists for c larger or smaller than a certain threshold, which varies according to the subinterval: we denote them by respectively.The expressions of these thresholds are not explicit, but we provide below rather precise estimates for them.We denote In the above pasting framework, c 0 involves the speeds of profiles connecting 0 with α and then α to γ, while c 1 refers to the connections γ to β and then β to 1.We denote with J the set of admissible speeds, i.e., the speeds c such that there is a profile with that speed satisfying (1.2).
The following main result concerns general necessary and sufficient conditions for the existence of wavefronts.
We point out that, in the case c 1 < c 0 , there are infinitely many profiles with the same speed c ∈ (c 1 , c 0 ).More precisely, see Figure 7, for every such c there exists λ c < 0 and a family of profiles ϕ λ , for λ ∈ [λ c , 0), which are characterized by for some ξ γ ∈ R. The first condition simply says that all profiles have been shifted so that they reach the value γ at the same ξ = ξ γ (in order to make a comparison possible); the second one states that their slopes at ξ γ cover the cone centered at (ξ γ , γ) and opening [λ c , 0).We refer to [2] for more information.
We denote the difference quotient of a scalar function of a real variable F = F (ϕ) with respect to a point ϕ 0 as We also introduce the integral mean of the difference quotient and denote it as Notice that for every ϕ ∈ (0, γ) there is ϕ 1 ∈ (0, γ) such that ∆(Dg, α)(ϕ) = δ(Dg, α)(ϕ 1 ).
Then we have sup and the same estimate holds true in the interval [γ, 1] by replacing α with β.
The following results provide necessary and sufficient conditions for the existence of decreasing profiles, in order to make condition (3.2) more explicit in terms of the functions f , D and g.The proofs are deferred to the end of this section.
In particular, wavefronts exist only if both conditions are satisfied.Notice that inequality (3.6) separates the behavior of f from that of Dg.
Remark 3.1.When f is strictly concave we have and then The following result shows, in particular, how far from zero must be the difference in (3.7) in order to have solutions.
Corollary 3.2 (Sufficient condition).We have the following results.
• We have (ii) either J ⊂ (−∞, 0) or J = ∅, in the case In the proof of Theorem 3.1 we will reduce the existence of a wavefront to equation (1.1) satisfying (1.2) to the investigation of a solution z to the following singular first-order problem in the interval [0, 1]: By a solution to (3.13) we mean a function z(ϕ) which is continuous on [0, 1] and satisfies the equation (3.13) 1 in integral form, i.e., Notice that we exploited here the assumption f (0) = 0.It is clear that such a z belongs to C 1 (0, 1) \ {α, β} .To solve problem (3.
Then we have: (a) For any c ∈ R there exists a unique where f (ϕ) : Conditions (3.15) also exploit estimates on the threshold speeds recently proposed in [12].With the help of Lemma 3.1, in the proof of the following proposition we analyze the subproblems we mentioned above.Proof.The proof analyzes the restriction of problem (3.13) to the four above intervals.
(3.21) Hence, inf We also denote w(ϕ) := −z(−ϕ + γ + β).Then in the interval [γ, β] problem (3.13) can be written as This concludes the analysis of the restrictions of problem (3.13) to the four above intervals.Condition c 1 ≤ c 0 is the requirement that there is a common admissible speed c for the above subproblems.In this case c ∈ [c 1 , c 0 ].
Remark 3.2.Since D and g vanish in the interior of none of the above sub-intervals, one finds ϕ ′ < 0 if ϕ ∈ (0, 1) \ {α, γ, β} (see [2, Proposition 3.1(ii)]).Moreover, by [4, Theorem 2.9 (i)], we deduce that the profile never reaches the value 1 for a finite value of ξ; the same result holds for the value 0, by exploiting again [4, Theorem 2.9 (i)] after the change of variables that led to (3.16).At last, we have ϕ ′ (γ) < 0 by the second part of the proof of Proposition 3.1(ii) in [2].As a consequence, the profile ϕ is strictly monotone.
Reasoning as there we obtain that lim hence ξ 2 β is a real value.With a similar reasoning, this time directly applied to z(ϕ 1,β ) and D(ϕ 1,β ), we can prove that also ξ 2 β is a real value.
The remaining pastings are exactly proved as in the proof of [2, Proposition 3.2] and we refer the reader to that paper for details.To this aim, in particular, we need that z(γ) > 0, which is satisfied when c 1 < c 0 by Proposition 3.1.The proof of the first statement is complete.
) and it satisfies (3.13).For more details we refer to the similar case presented in the proof of [2, Proposition 3.1 (ii)], which applies because g satisfies [2, (2.2)].According to Proposition 3.1 we obtain c 1 ≤ c 0 and then also the second statement is proved.
Remark 3.3.We now provide a simple argument showing why wavefronts should exist for suitable concave f , in the case the drift ḟ is first positive and then negative.For λ > 0, let f be defined by λu in (0, γ) and −λ(u − 2γ) in (γ, 1), so that f is Lipschitz continuous with ḟ = λ in (0, γ) and ḟ = −λ in (γ, 1).In this case, the role of λ is to shift to the right (of magnitude +λ) the estimates for c 0 , as (3.19) and (3.22) show, and to shift to the left (of −λ) the estimates for c 1 (see (3.24) and (3.26)).Hence, (3.2) holds true for λ large enough.
We now investigate when the set J of admissible speeds contains positive values.

Existence of wavefronts in the model of biased movements
In this section we investigate the presence of wavefronts to the biased model (2.9) and prove their main qualitative properties.We make use of the results provided in Section 3 for a general reaction-diffusion-convection process.
, and then v vanishes at the maximum density 1; it can also possibly vanish at u 0 = p+q p (i.e., if u 0 ∈ [0, 1)).This is analogous to similar models in collective movements [21, §3.1].Recalling that q < 0, it is easy to see that only the following cases may occur (for simplicity we do not include the case p + q = 0, when u 0 = 0, or p = 0, when u 0 is missing, for which slightly different results hold): 1. q < 0 < p+q.Then v is concave, it is first negative, then positive; f is convex-concave.
2. p + q < 0 < p. Then v is positive and concave; f is concave or convex-concave.
A sufficient condition for the existence of wavefronts to equation (2.9) is (3.10).The following result provides an upper estimate of the right-hand side of (3.10).Lemma 4.1.We have Then, for ϕ ∈ [0, γ] we obtain By (3.4) and (4.7) we deduce 2 sup With a similar reasoning we have that 2 sup since we have We complete the proof by combining (4.8) and (4.9).

A strictly concave convective term
The left-hand side of (3.10) takes a simple form when f is strictly concave (see Remark 3.1); for this reason we first consider this case, see Figure 8.The following result characterizes the strict concavity of the function f , see (2.21).Proof.By (4.2) we compute f (u) = −6pu + 4p + 2q; therefore f < 0 in (0, 1) if and only if − 3pu + 2p + q < 0, for any u ∈ (0, 1).(4.10) The line −3pu + 2p + q = 0 connects the points (0, 2p + q) and (1, −p + q).We remark that 2p + q = −p + q = 0 is not possible since q < 0 by conditions (2. ≤ C i D i + C g D g ≤ C g D g ; this condition does not match with the assumption C g < 0, which is necessary to have wavefronts to equation (2.9) satisfying (1.2) by Proposition 2.1.Indeed, the bare convexity of f in [α, β] is sufficient to hinder the existence of such wavefronts, because the right-hand side of (3.6) is strictly positive when D and g are as in (4.3), (4.4).
We now apply the sufficient condition (3.10) to the current case.
Proof.In order to apply (3.10) we exploit Remark 3.1.Then, by exploiting (4.5), we compute By (4.1) and (2.14) we can write Therefore, when f is strictly concave we have inf By the above formula, Lemma 4. We now investigate the sign of the speed of wavefronts; this issue is important in the biological framework.We find below conditions in order that wavefronts with positive speed exists and conditions assuring that every wavefront has negative speed.

A convective term which changes concavity
We now consider a convective term f as in (2.10) (see also (4.2)) which changes its concavity in [0, 1] and show that also in this case the model (2.9) can support wavefronts satisfying condition (1.2).Due to the definition of f , a concavity change occurs iff p = 0, and in this case only once, namely at 2  3 + q 3p .Moreover, when this occurs, then concavity and convexity are strict.Lemma 4.4.Assume that f has an inflection point in (0, 1).Then: (i) f is first convex and then concave if and only if sd > 3  2 .
(ii) f is first concave and then convex if and only if s < 0.
To simplify calculations, in the following we only consider the case when γ, which is the inner zero of g and is given by (2.14), coincides with the inflection point of f ; i.e., we assume in the current section (without further mention)

The convex-concave case
We consider a function f which is first convex and then concave, with γ as inflection point, see Figure 5.
We recall that we are assuming γ ∈ (α, β), see (2.17).We now check the implications of this condition on sd, because γ also satisfies (4.21).By Lemma (4.4)(i) and (2.15) 1 we obtain that C i D i + C g D g > 0 and hence γ < 2 3 < β by (4.21).On the other hand, the condition γ > α is equivalent to sd > 1 + 1 ω > 2 because of (2.13) and (4.21), which strengthens the previous requirement sd > 3  2 .Summing up, under the assumptions of the current case, the parameters sd and γ must satisfy the conditions (4.23) We now consider the issue of the existence of profiles.By making use of (4.5), the left-hand side of (3.10) becomes inf where We now investigate the sign of (4.24): its positivity is necessary for (3.10) to hold.The set S has been defined in (2.25).
The following result shows the existence of wavefronts for equation (2.9) satisfying (1.2) when f is first convex and then concave.

Figure 1 :
Figure 1: Typical plots of the functions D (dashed line) and g (dashdotted line).

Figure 2 :
Figure 2: Sketch of the meaning of the parameters a i,g and b i,g .

Figure 3 :
Figure 3: On the left: the triangle T g (d) is the intersection of three half-planes.Here |C g | D g = 6; dashed lines refer to d = 5, solid lines to d = 8.On the right: The conditions in (2.20) prescribe that µ must lie between the red and the blue line.Condition (2.22) further prescribes that µ > 0 must belong to the region below the black line: the dashed curve refers to E g = 40, the dash-dotted curve to E g = 60.

Figure 4 :
Figure 4: Thresholds leading to wavefronts with positive or negative speeds when f is strictly concave.The figure is to be interpreted as follows: if d > 4 + 2 √ 3 ∼ 7.46, then there exist wavefronts with positive speed, while, if 4 < d < (5 + 2 √ 3)/2 ∼ 4.23, then every wavefront has negative speed under the assumptions of Theorem 4.3.

Figure 5 :
Figure 5: Plots of the functions D (dashed line), g (dashdotted line) and f (solid line) in the case f is convex-concave (on the left) and concave-convex (on the right), with γ as inflection point of f .

Figure 6 :
Figure 6: The sets S, on the left, and S, on the right.The sets S and S are the plane regions bounded from above by the blu curve and from below by the red curve.

Figure 7 :
Figure 7: Some profiles with the same speed c, in the case c 1 < c 0 in Theorem 3.1; profiles have been shifted so that ϕ(ξ γ ) = γ.

Figure 8 :Lemma 4 . 2 .
Figure 8: Plots of the functions D (dashed line), g (dashdotted line) and f (solid line) in the case f is strictly concave.

Theorem 4 . 1 .
If f is strictly concave and d − 1

Figure 10 :
Figure 10: The triangle T g (d) (thick black lines) and the cone R(d), which is bounded from below by the red line and from above by a black line.Here |C g | D g = 6 and d = 10.
In turn, this condition is satisfied if (4.14) holds.