Asymptotic harvesting

Asymptotic harvesting of populations in random environments

Alexandru Hening Department of Mathematics
Tufts University
Bromfield-Pearson Hall
503 Boston Avenue
Medford, MA 02155
United States
alexandru.hening@tufts.edu
Dang H. Nguyen Department of Mathematics
Wayne State University
Detroit, MI 48202
United States
dangnh.maths@gmail.com
Sergiu C. Ungureanu Department of Economics
City, University of London
EC1V 0HB
United Kingdom
Sergiu.Ungureanu.1@city.ac.uk
 and  Tak Kwong Wong Department of Mathematics
The University of Hong Kong
Pokfulam
Hong Kong
takkwong@maths.hku.hk
Abstract.

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction – instead the harvested population converges to a unique invariant probability measure.

When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate.

We also show that, if the yield function is and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

Key words and phrases:
Ergodic control; stochastic harvesting; ergodicity; stochastic logistic model; stochastic environment
2010 Mathematics Subject Classification:
92D25, 60J70, 60J60
D. Nguyen was in part supported by the National Science Foundation under grant DMS-1207667.
T. K. Wong was in part supported by the HKU Seed Fund for Basic Research under the project code 201702159009, and the Start-up Allowance for Croucher Award Recipients.

1. Introduction

Many species of animals like whales, elephant seals, bisons and rhinoceroses, are at risk of being harvested to extinction ([Gul71, RPLB81, LHW93, Pri06]). Excessive harvesting has already led to both local and gobal extinctions of species ([LES95]). In fact, a significant percentage of the endangered birds and mammals of the world are threatened by harvesting, hunting or other types of overexploitation ([LES95]), and there are similar problems for many species of fish ([HR04]). This is why harvesting strategies have to be carefully chosen. After significant harvests, it takes time for the harvested population to get back to the pre-existing level. Moreover, the harvested population fluctuates randomly in time due to environmental stochasticity. As a result, an overestimation of the ability of the population to rebound can lead the harvester to overharvest the population to extinction ([LES95]). A less common but nevertheless important problem is an insufficient rate of harvesting. Because of instraspecific competition, the population is bounded in a specific environment, so an extraction rate that is too low would lead to a loss of precious resources. For the same reason, choosing an efficient extraction strategy for valuable species is important ([Kok01]).

We present a stochastic model of population harvesting and find the optimal harvesting strategy that mazimizes the asmptotic yield of harvested individuals. We consider a novel framework, the one of optimal ergodic harvesting. This is based on the theory of ergodic control ([ABG12]). In most stochastic models that exist in the literature, for example [LES95, AS98, LØ97], the population is either assumed to become extinct in finite time, or it can end up being harvested to extinction. In our framework, if the population goes extinct under some harvesting strategy, the asymptotic yield is and therefore this strategy cannot be optimal. If one wants to ensure that harvested species are preserved, this framework is a natural candidate. Our aim is to present a theory of optimal harvesting that includes the risks of extinction from both environmental noise and harvesting. We assume that the population is homogeneous and can be described by a one dimensional diffusion. The harvesting rate is assumed to be bounded, as infinite harvesting rates would imply an unlimited harvesting capacity, something that is clearly not realistic.

In most cases, environmental noise can be introduced in the system by transforming differential equations into stochastic differential equations (SDE). Such techniques require dealing with significant mathematical difficulties, but their use is not just a case of honoring generality. First, there are direct effects of stochasticity on the predictions of the model, and the parameters quantifying it show up in the results. Second, any realistic biological system will depend on environmental variables that are not, or cannot be, accounted for. The role of stochasticity is to ensure that the solutions proposed are robust to such omissions. For example, if avoiding extinction is important, deterministic models can give misleading solutions even when their parameters are corrected for noise ([Smi78]). The transformation to SDE works especially well when the environmental fluctuations are small and there is no chaos ([LES95]). We focus on models with environmental stochasticity and neglect the demographic stochasticity which arises from the randomness of birth and death rates of each indiviual of a population. Throughout the paper we assume that environmental stochasticity mainly affects the growth rate of the population (see [Tur77, BM77, MBHS78, Lei81, Bra02, Gar88, EHS15, ERSS13, SBA11, HN18a] for more details). For computational tractability and for clarity of exposition, we look at a one-dimensional model. Nevertheless, our framework works for any model that can be written as a system of stochastic differential equations (satisfying some mild assumptions - see [ABG12]).

A major limitation of existing models in the literature is the dependence of the optimal solutions on parameters that are hard to quantify. For example, in [LES95] the level at which the population becomes extinct – the minimal viable population – must be assumed; without it the yields become infinite. In [AS98] the yield must be time discounted to avoid maximizing over yield infinities, and this requires providing a time value for resources. The minimal viable population is a difficult scientific question ([Sha81, TBB07]), and the time value of yields is a difficult economics and policy question, because it implies the comparison of the utility of present and future generations ([DS87]). In contrast, our model sidesteps the issue by assuming no time preference – and therefore no bias towards extracting in the present, and resolves the problem of maximizing over infinite yields naturally by looking at asymptotic behavior.

A particular case of our model was studied in [AP81]111We thank the anonymous referee who has brought the paper to our attention.. The authors limited themselves to the analysis of harvesting strategies that were of bang-bang type. In [Aba79], one of the co-authors in [AP81] proved that an optimal gathering strategy was necessarily of a bang-bang type in a continuous time Markov chain model, making use of the simplifying assumption of a finite state space. Here, instead, we look at very general possible harvesting strategies in a continuous state stochastic model, and show that the optimal one is of bang-bang type. Our contribution is therefore two-fold. We generalize the setting of [AP81] significantly by looking at very general density-dependent growth rates, not just the logistic case. Moreover, we prove what the authors of [AP81] intuited, namely that the optimal strategy is of bang-bang type; and furthermore that this is true for the larger class of convex yield functions.

Stochastic optimal control applications are common in the finance literature. Following the seminal contributions of [Mer69, Mer71], objective functions that are integrals of time discounted instantaneous utility flows are now standard. The crucial simplifying assumption is that of time-additive total utility. The utility flow usually depends on consumption flows, and therefore indirectly on other variables and stochastic constraints. With the time-additive utility assumption, our general yield function can also be interpreted as an instantaneous utility function dependent on yield, and our objective function can be the asymptotic expected utility flow dependent on yield. Because a population stock cannot grow indefinitely in our biological model, we diverge from the general finance literature, where financial returns do not usually depend on the size of the holdings of an individual.

Finally, we generalise a result from one of the stochastic models in [Smi78], where the equivalent to our yield function has a specific simple form. We show that, when the yield function is weakly convex, the optimal control is bang-bang. However, if the yield function is strictly concave, then the optimal harvesting strategy has to be continuous, in contrast to the bang-bang type optimal strategy we find for a linear yield function. This generalization is useful for economic welfare analysis (a more general form of cost-benefit analysis), which typically relies on a concave utility function, equivalent to the concave yield function herein. In economic models, concavity is assumed to model risk aversion (see [MCWG95, Proposition 6.C.1] for justification), and for the convenience of interior solutions to maximisation problems. Concave utility leads to a trade-off between risk and returns in asset choice [Mer71], so the connection between yield concavity and strategy continuity mentioned above is suggestive of risk management. However, risk management interpretations from the finance literature are not directly applicable here. First, financial asset returns are assumed reasonably to not be decreasing in the asset value owned by investors.222The assumption may not apply in models with large institutional investors. Moreover, the risk-return trade-off is captured in models with choice between at least two assets with different risk profiles.333An ecological model extension that would link this literature to our model would consider optimal extraction policy to maximise a time discounted concave total-yield function when there are at least two populations, situated in different environments with no growth limitation. If anything, finding a bang-bang optimal strategy when yield is linear is more related to finding corner solutions in maximisation problems with linear utility. A bang-bang strategy uses one of the two extremes of the harvesting rate, depending on the momentary population stock.

The rest of the paper is organized as follows. In Section 2 we introduce our model and results. We prove that, if the population in the absence of harvesting survives, the yield function is the identity and the harvesting rate is bounded above by some number , then the optimal strategy is always a bang-bang type solution: there exists an such that one does not harvest if the current population size lies in the interval and harvests at the maximal possible rate, , if the current population size lies in the interval . The proofs of the above results are collected in Appendix A. In Section 4 we apply our results to the special setting of the logistic Verhulst-Pearl model. In Section 3 (proofs in Appendix B) we show that if the yield function is strictly concave, the optimal harvesting strategy is continuous, and when the yield function is more generally weakly convex, optimal strategy is bang-bang.

Finally, in Section 5 we offer some numerical simulations that show how the optimal harvesting strategies and optimal asymptotic change with respect to the parameters of the model. We also provide a discussion of our results.

2. Optimal ergodic harvesting

We consider a population whose density at time , in the absence of harvesting, follows the stochastic differential equation (SDE)

(2.1)

where is a standard one dimensional Brownian motion. This describes a population with per-capita growth rate given by when the density is . The infinitesimal variance of fluctuations in the per-capita growth rate is given by .

The following is a standing assumption throughout the paper.

Assumption 2.1.

The function satisfies:

  • is locally Lipschitz.

  • is decreasing.

  • As we have .

  • The function has a unique maximum.

The behavior of (2.1) is not hard to study. In the particular case when see [EHS15, DP84]. The methods there can be easily adapted to our setting. Alternatively, one could use the general results from [HN18a]. The process does not reach or in finite time and the stochastic growth rate determines the long-term behavior in the following way:

  • If and , then converges weakly to its unique invariant probability measure on .

  • If and , then almost surely.

We let and throughout the paper.

Assume that the population is harvested at time at the stochastic rate for some fixed . Adding the harvesting to (2.1) yields the SDE

(2.2)

A stochastic process taking values in is said to be an admissible strategy if is adapted to the filtration generated by the Brownian motion . Let be the class of admissible strategies. An important subset of is the class of stationary Markov strategies, that is, admissible strategies of the form where is a measurable function. By abuse of terminology, we often refer to the map as the stationary Markov strategy. Using a stationary Markov strategy , (2.2) becomes

(2.3)
Remark 2.1.

The sigma algebra gives one the information available from time to time . An admissible harvesting strategy is therefore a strategy which can take into account all the information from the start of the harvesting to the present. These strategies are much more general than constant strategies. Stationary Markov strategies are the harvesting strategies which only depend on the present state of the population density.

We associate with the family of generators defined by their action on functions with compact support in as

(2.4)

We will call a yield function if the following assumption holds.

Assumption 2.2.

The function satisfies:

  • is continuous.

  • .

  • has subpolynomial growth that is, there is such that for .

Our aim is to find the optimal strategy that almost surely maximizes the asymptotic yield

(2.5)

In other words we want to find such that, for any initial population size , we have with probability that

We note that many of the existing models that look at the optimal harvesting of a population in a stochastic environment ([LØ97, AS98, LES95]) assume that the yield function is the identity i.e. . This assumption is not always justifiable (see [Alv00]) and as such we present in Section 3 results for more general functions .

Remark 2.2.

We note that if has an invariant probability measure on , then for any almost surely

In particular, if goes extinct, that is, for any we have with probability

then the only invariant ergodic measure of on is the point mass at , and hence, we get that with probability

Our method for maximizing the asymptotic yield forces the optimal harvesting to be such that the population persists.

Remark 2.3.

By [ABG12, Theorems 2.2.2 and 2.2.12], the controlled systems (2.2) and (2.3) have unique local solutions on for any admissible control and stationary Markov control respectively. Note that one can find large enough such that

With this fact in hand, we can use the arguments from [Kha12, Theorem 3.5] to obtain the existence of global solutions on of (2.2) and (2.3). In particular we get that

The main result of the paper is the following. \thmt@toks\thmt@toks Assume that and that the population survives in the absence of harvesting, that is . Furthermore assume that the drift function satisfies Assumption 2.1. The optimal control (the optimal harvesting strategy) has the bang-bang form

(2.6)

for some . Furthermore, we have the following upper bound for the optimal asymptotic yield

(2.7)

3. Continuous vs bang-bang optimal harvesting strategies

As showcased in Theorem 2, when is the identity function the optimal harvesting strategy is of bang-bang type. In Appendix B we prove the following result.

\thmt@toks\thmt@toks

Suppose Assumption 2.1 holds and the yield function satisfies

  1. ,

  2. is strictly concave.

\thmt@toks
\thmt@toks

Then the optimal harvesting strategy is continuous and given by

Furthermore, the HJB equation for the system becomes

(3.1)
Remark 3.1.

We cannot find the exact form of the optimal harvesting strategies in this case. Note that in Theorem 2 we have which is not strictly concave nor strictly convex.

Intuitively, this is not unlike maximising a strictly concave objective function under a linear constraint. The optimal choice usually moves smoothly over the domain as the direction of the constraint changes. However, when the objective function is weakly convex, e.g. linear, the optimum will jump on the allowed interval.
Here, we show that if the yield function is weakly convex, the optimal control is bang-bang. The optimal strategy has a similar form to the one for linear yield, if a further assumption on the joint rates of change of the population growth rate and the yield function is made. \thmt@toks\thmt@toks Assume that is weakly convex, grows at most polynomially, and the population survives in the absence of harvesting, that is . Furthermore assume that the drift function satisfies the following modification of Assumption 2.1:

  • is locally Lipschitz.

  • is decreasing.

  • As we have .

  • The function

    (3.2)
    \thmt@toks
\thmt@toks

has a unique extreme point in which is a minimum. \thmt@toks

\thmt@toks

If the assumptions (i)-(iii) hold, the optimal control has a bang-bang form (i.e., the harvesting rate is either 0 or the maximal ). If assumptions (i)-(iv) hold, the optimal harvesting strategy has a bang-bang form with one threshold

for some .

4. The logistic case:

Throughout this section we provide a thorough analysis of the logistic Verhulst-Pearl model. As such, we will assume that the growth rate is for positive constants . It is clear that this satisfies Assumption 2.1. If we harvest according to a constant strategy then the SDE (2.3) becomes

It is then easy to see that, as long as , the asymptotic yield is

We can maximize this yield , which is quadratic in . The maximum will be at

and the maximal asymptotic yield (among constant harvesting strategies) is

Note that is also called maximum sustainable yield (MSY) in the literature. Since we note that

Combining this with (2.7) one sees that the optimal asymptotic yield satisfies

Note that Theorem 2 does not give us information about , the point at which one starts harvesting.

One possible strategy to find out more information about is the following: Look at controls of bang-bang type that have a threshold at and then maximize over all possible . This will then give us a way of finding . Let be the harvesting strategy

(4.1)

For this control our diffusion (2.3) (with ) is of the form

(4.2)

for

and

Standard diffusion theory shows (see [HK18, BS12]) that the boundary is natural and the boundary is entrance for the process from (4.2). As a result, when , one can show using [BS12] that the density of the invariant measure is of the form

(4.3)

where is a normalizing constant given by

In this case the harvesting yield is

(4.4)
Figure 1. Typical shape of the asymptotic yield function as a function of the harvesting threshold , where one begins to harvest. Here for and .

By Theorem 2 the point has to satisfy:

It is clear that is differentiable, that exists and satisfies . Therefore, is a solution of

(4.5)

The condition above can be restated as an equation involving incomplete gamma functions. We were not able to prove analytically that (4.5) has a unique solution. [BP06, BP08] show possible analytical methods that can be applied to such equations in a simple case. However, numerical experiments that we have done support this conjecture (see Figure 1).

Conjecture 4.1.

There exists a unique such that . Furthermore, the optimal harvesting strategy is given by

5. Discussion and future research

We have analysed a population whose dynamics evolves according to generalization of the logistic Verhulst-Pearl model in a stochastic environment, but subjected to strategic harvesting. The rate at which the population gets harvested is bounded above by a constant , and the harvested infinitesimal amount is proportional to the current size of the population. We show that the harvesting strategy , which describes the harvesting rate and is chosen to maximize the asymptotic harvesting yield

is of bang-bang type, i.e. there exists such that

Figure 2. Graph of the maximal asymptotic yield as a function of the harvesting threshold for different values of the growth rate . We take , , and (blue), (orange), (green), (red), and (purple).

5.1. Logistic Verhulst-Pearl

In the particular case when , we can give more information about as follows: The harvesting yield function is determined, by letting the jump in the bang-bang control be at . That is, we look at the yield when the control is

The typical behavior of the point where is maximized and of as the parameters and change was analyzed numerically and is presented in Figures 2, 3 and 4, with the normalization .

We note from numerical experiments that increasing the growth rate increases the threshold at which one should start optimally harvesting (Figure 2). This is an intuitive result, since an increased growth rate increases the maximal equilibrium value of the population in the equivalent deterministic growth model with competition ([Smi78]). Therefore, it should also increase asymptotic harvesting yield, as well as the point at which harvesting should start. Moreover, higher growth rates make the population get faster to the point where one starts harvesting, reducing the cost of a delay.

If one increases the maximal harvesting rate then the harvesting threshold is also increased (Figure 3). This also makes sense because if , then a population with constant harvesting rate will go extinct almost surely. An increase in the harvesting threshold is necessary to make sure that there is no extinction. Moreover, as gets larger one can wait longer to start harvesting. With a larger maximal rate available, there is less chance that there will be losses because the population overshoots the optimal extraction point. Similarly, increasing the harvesting rate also increases the maximal asymptotic harvesting yield, for the obvious reason that there is better control on the population level and therefore extraction can happen closer to the optimal level.

In contrast, if one increases the intraspecific competition rate , then the harvesting threshold decreases (Figure 4). The equilibrium value of the population in the equivalent deterministic model ([Smi78]) decreases with , and as a result so does the extraction rate. Evidently, even in the stochastic model, if competition is very strong the population cannot spend much time at high densities, and therefore one has to start harvesting early. An increase in will also decrease the maximal asymptotic harvesting yield.

Figure 3. Graph of the maximal asymptotic yield as a function of the harvesting threshold for different values of the maximal harvesting rate . We take , , and (blue), (orange), (green), (red), (purple), and (brown).

We are able to prove that the maximal asymptotic yield satisfies the inequality

In particular, the bang-bang optimal strategy has a higher asymptotic yield than the optimal constant harvesting strategy. Moreover, the bang-bang optimal strategy gives a lower asymptotic yield than the optimal constant harvesting strategy in the absence of noise. This means that the analysis of the more complex stochastic model was fruitful, recommending a qualitatively different strategy. Moreover, environmental fluctuations decrease the maximal asymptotic yield and, because the correction is negative, protecting a population from extinction requires a careful measurement of natural fluctuations when designing optimal harvesting. When environmental stochasticity was not taken into account, harvesting often lead populations to extinction ([LES95]).

Real populations do not evolve in isolation. As a result, ecology is concerned with understanding the characteristics that allow species to coexist. Harvesting can disturb the coexistence of species. In future research we intend to tackle multi-dimensional analogues of the setting treated in the current article. Natural models for which one can add harvesting would be predator-prey food chains ([GH79, Gar84, HN18b, HN18c, TL16]), more general Kolmogorov systems ([SBA11, HN18a]) and structured populations where there can be asymmetric harvesting ([ERSS13, EHS15, HNY18, RS14, BS09, SR11]). In the multi-dimensional setting the Hamilton-Jacobi-Bellman (HJB) equation becomes a PDE and the analysis becomes significantly more complex. New tools will have to be developed to tackle these problems.

Figure 4. Graph of the maximal asymptotic yield as a function of the harvesting threshold for different values of the intra-competition rate. We take , , and (blue), (orange), (green), (red), and (purple).

Above we have imposed a bound on the extraction rate, . This was because it is a realistic feature, but it was also practical for the analysis. Nevertheless, it is interesting to consider the case when the extraction rate is unbounded. A practical model with no extraction limit corresponds to having unlimited control over a target population, which is sometimes the case. Such a model would have the benefit of not requiring a nuisance parameter that may be hard to determine.

5.2. Concave and convex yield functions

We have also studied the more general case involving concave and convex yield functions. When the yield function is strictly concave, it was shown that the optimal control is not bang-bang, but continuous in the population parameter. Vice-versa, when the yield function is weakly convex, we have shown that the optimal control is necessarily bang-bang. Moreover, if a certain further assumption on the relative rate of growth of and holds, we can also show that the bang-bang optimal control has a single threshold where the extraction rate goes from 0 to – as in the linear special case.
This generalization allows us to think of applications of population harvesting where the yield function is in fact a utility function, or some other more general social welfare measure.

5.3. Unbounded harvesting

If we allow for general, possibly unbounded, harvesting we would have to study the Skorokhod SDE

(5.1)

where is supposed to be non-negative, increasing, right-continuous and adapted to - we denote the set of all such strategies by . Then the problem is to maximize the asymptotic yield, i.e. find

We want to find the harvesting strategy , which we call the optimal harvesting strategy, such that

The analysis above, for the bounded harvesting rate, determined that the optimal strategy has a bang-bang property, where extraction is maximal after some cut-off. This suggests that raising the maximum would not change the bang-bang property, but determining that result required a bounded extraction rate. Thinking of the limiting behavior of the yield function above shows the difficulty: as , the density of the distribution above the cut-off goes to 0 (see (4.3)). The conjectured optimal solution is akin to having a reflective boundary at , and the yield is determined by the time spent close to the boundary.

Conjecture 5.1.

Assume that the population survives in the absence of harvesting i.e. . The optimal extraction strategy has the form

(5.2)

for some , where is the local time at of the process from (5.1).

This conjecture is supported by the results from [AS98] where the authors study the maximization of the discounted yield

and is the extinction time. It is shown in [AS98] that the optimal harvesting strategy is of the form (5.2). One possible approach to prove Conjecture 5.1 would be to use the results from [AS98] and then let the discount factor go to .

Acknowledgements. We thank two anonymous referees for very insightful comments and suggestions that led to major improvements.

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Appendix A Proofs

In this appendix we present the framework of ergodic optimal control and prove the main results of our paper.

For any , denote the unique invariant probability measure of on by if it exists. Define

Let . Since , there exist constants such that

(A.1)

By Dynkin’s formula

Thus,

(A.2)

As a result, the family of occupation measures

is tight. If has an invariant probability measure on , then converges weakly to because the diffusion is nondegenerate. This convergence and the uniform integrability (A.2) imply that

If has no invariant probability measures on , then the Dirac measure with mass at is the only invariant probability measure of on . Moreover, any weak-limit of as is an invariant probability measure of ([EK09, Theorem 9.9] or [EHS15, Proposition 8.4]). Thus, converges weakly to the Dirac measure as . Because of (A.2) and , we have

Thus, we always have

(A.3)

Define

(A.4)

It will be shown later that whenever the population without harvesting persists, i.e. when .

Theorem A.1.

Suppose , satisfies Assumption 2.1 and satisfies Assumption 2.2. There exists a stationary Markov strategy such that exists and . Moreover, for any admissible control , we have

Proof.

By (A.2) and since has a subpolynomial growth rate we can conclude that

(A.5)

Moreover, since we note that, since our population does not go extinct, . On the other hand, since is continuous and we get that for is sufficiently small. This fact combined with (A.5) implies the existence of an optimal Markov strategy according to [ABG12, Theorem 3.4.5, Theorem 3.4.7]. ∎

Theorem A.2.

Suppose , satisfies Assumption 2.1 and satisfies Assumption 2.2. The HJB equation

(A.6)

admits a classical solution satisfying and . The solution of (A.6) has the following properties:

  • For any

    (A.7)
  • The function is increasing, that is

    (A.8)

A Markov control is optimal if and only if it satisfies

(A.9)

almost everywhere in .

Proof.

Consider the optimal problem with the yield function

for some fixed and . Note that this is the -discounted optimal problem. Pick any and let , be the solutions to the controlled diffusion

with initial values , respectively. Note that we are using a fixed admissible control which is the same for any initial value. The control here is not a Markov control which in general depends on the initial value. Since is continuous and decreasing, for , there exists depending continuously on such that . Using Itô’s Lemma we have

which in turn yields

Therefore, if , we get that

This implies that is an increasing function. Therefore, the optimal yield

is also increasing. By [ABG12, Lemma 3.7.8], there is a function satisfying (A.6) for a number such that

(A.10)

Moreover,

for some sequence that satisfies as . This implies that is an increasing function, i.e.

(A.11)

For any continuous function satisfying

(A.12)

we have from (A.1) and [ABG12, Lemma 3.7.2] that exists and satisfies

(A.13)

and

(A.14)

where Moreover, by using [ABG12, Lemma 3.7.2] again we get that

(A.15)

where

and

By [ABG12, Formula 3.7.48], we have the estimate

which implies

(A.16)

Now, pick any and divide (A.16) on both sides by . We get

and by letting