Optimal Harvest Problem for Fish Population—Structural Stabilization
Round 1
Reviewer 1 Report
The present manuscript studies the optimal harvest problem addressed within the continuous model of temporal and either age or size dynamics of a population with environmental effects in recruitment. The authors formulate the optimization system including the general model of population dynamics and the profit optimization problem, and study the questions of existence and uniqueness of its solution. Then the authors formulate three special cases of general model: the model of 1) temporal dynamics, 2) temporal and age dynamics, 3) temporal and size dynamics of a population. For the latter model and without diffusion term, the authors provide more in-depth analysis, in particular, deriving the stationary solution and studying its properties. Finally, the authors present the results of numerical study of the model with size structure, considering scenarios with and without fishing. Overall, the manuscript presents an interesting and original study. Its results will certainly be of great interest to the broad community of mathematical ecology and in particular to fisheries scientists. However, the manuscript in its present form requires a major revision. Please see comments and suggestion for improvements below.
1. Major comments
1.1. The introduction provides a very general context, pointing to several gaps in existing research and observations, but it’s missing the point on how this work fits into (and ideally expands upon) the current knowledge and modeling practices. Please add a paragraph. The manuscript will also benefit from revision to make it more focused. What is the central question? For example, despite being one of the focal points of the manuscript, results on stabilization under the influence of the size structure are practically mentioned only in the abstract. The results section does not have corresponding statement, nor conclusion or discussion. Please elaborate what is meant by stabilization, how it was measured, how it may change the fisheries management, and provide some arguments on the role of size structure as well as fisheries in the stabilization of population dynamics.
1.2. The structure of the manuscript has some room for improvement. The manuscript is structured in four parts: 1) Introduction, 2) general model, 3) modeling of population 4) Fishery model and 5) Results and Discussion. First, sections 3 and 4 can be combined as they both deal with the special cases of system (1-3). Second, the section on the age-structured models is limited to the formulation of the model with its optimal harvest problem. Is it important to keep it in the manuscript given the fact that the problem is not studied further?
On the other hand, it is desirable to have a separate section devoted to the numerical solution of model (1-2) and optimal harvest problem (3). It is bit confusing that the numerical methods are mentioned just after the general model, but the actual numerical experiments were done for the model special cases with size structure. Actually, starting 4.2. the manuscript describes numerical settings and simulations. It would be more logical to talk about numerical methods there.
Finally, please separate the Results and Discussion. Blended with the Results and limited to only scarce notes, the discussion is actually missing. The discussion is important to explain results and to highlight the value and novelty of the results. Here are some suggestions for consideration. What are the limitations of each model and applicability? Do the analytical properties of optimization system in part 2 hold for the spatiotemporal models as the formulation of model (1-2) is more often used for x being a spatial variable? Are there cases when the model with age structure (also formulated in the paper) should be preferred to the size structure? And of course, the stabilization of population dynamics must be discussed.
1.3. It is understandable that the continuous model is formulated in the most general form. However, I have some reservations as to the use of operators in model (1-2) for the case when x denotes space. Namely, the defined derivative operator becomes gradient and \nabla \cdot becomes divergence operator. Applied to the scalar y, divergence is not defined (page 3, last paragraph) as it only makes sense to apply it to a vector field. In this case the diffusion term should be written as \nabla \cdot (k \nabla y), that is div grad. Please clarify also that \delta(y) is not a Laplacian, but the diffusive term with diffusion rate being space and time dependent.
Also, what is the interpretation of diffusion term in case of size-structured model? Since diffusion process transports the density in both directions around x, it is not intuitive at all to grasp its mechanism and underlying biological process. It is mentioned later (page 13) that it is related to the dispersion of growth rate, but dispersion of growth rate is not the same as diffusion of density over size, so the explanation remains vague.
1.4. There are multiple parameters in the model with optimization problem (6), with “local” change function described by (19) and the profit function (20). Since section 4.3 describes only the analytical relationships and simplifications being made, it is impossible to derive parameters that were used in the numerical experiments and led to the results presented in the manuscript. Unless, any set of parameter values, which obey the functional relationships in 4.3 will give the same results, and if it’s the case, it should be explicitly stated, the parameter values used in numerical experiments as well as their units should be listed in a Table.
1.5. The manuscript clarity and English need improvements. The good point is that all sentences are short and, as such, ought to be clear. However the peculiar use of English words or redirection to some earlier ideas with indefinite pronouns is confusing. Sometimes it is difficult to follow simply because logical connections are missing. See examples in minor comments.
2. Minor comments
Page 1. “Parameters… obtained from this procedure”. Please be more specific? From previous sentence it is not clear what procedure is meant.
Page 1. “high-gradient dynamics” refers here (and in a several other places) to temporal dynamics, which is confusing as gradient is defined as the derivative over space and it’s a vector. Instead, it can be said something like “harvesting intensity dynamics is characterized by high rates” or similar.
Page 2. It is not catching the mutliple species that is the serious problem, but the quality of reporting of catch by species.
Page 2. “Various fishery strategies have been developed to fishery theory”. Not clear, needs to be rewritten.
Page 2. Last paragraph, three sentences starting with “Models… . Models… . Modelling … .”. If it is important to mention, then please be more specific while referring to particular models and problems.
Page 3. “The fundamental model … is a pattern for MSY models”. Not clear.
Page 3. “The structure is a stable factor…”. Did you want to say “a stabilizing” factor? Otherwise, the structure is also stable?
Page 3. “The set D describe … spatial arrangement”. Spatial domain?
Page 3. “All functions in ratios” modify to “All functions in model (1) with initial and boundary conditions (2)”, or, in more concise form “all functions in model (1-2)”.
Page 4. Please define the term ‘harvesting intensity’ here so that the reader does not have to look for the use of this function in the model in order to understand what it is. The term “harvesting intensity” is not widely used in nowadays literature, rather there are more field terms such as catch, catch rate (catch per unit of effort), fishing effort, fishing mortality rate. From the use of function u it is a fishing mortality rate. By the way, later on in part 3 you call a function h, which is instantaneous catch, the harvesting intensity (page 6). Please correct.
Page 4. Modify “the practicality of a harvest” to “The function of profit in a fishing industry” or “The profit for the fishery”, or “the profitability of a fishery”, if the latter is more exact.
Page 4. Modify “basic model” to “general model”. Same on page 6.
Page 5. Modify “target functional” to “cost function”.
Page 6. “Using (9) we obtain”. Do you mean (T9)?
Page 6. “have the next forms” --> “have the following forms”
Page 7. “an increase of biomass is telling as function” --> “… is described by function”
Page 7. “concrete situations” --> “special cases”
Page 7. “parameters u_a are defined additivity from other formulas”. The sentence needs to be revised as it is not clear what the well-known result is.
Page 7. At the beginning the text says “we can investigate some special cases”, but the case 2 does not seem to include any investigation. Perhaps, it should be “Below are the possible definitions of fishing mortality rate in system (9)”
Page 8. “A depend recruitments from parents’ quantity is absent for this describing”. Suggestion: “It is assumed that there is no stock-recruitment relationship in the population”.
Page 9. “Parameter gamma describes the nonlinear…”. How is this parameter defined? It is only later (page 10) that the reader can find that gamma>1, but is it always so? Please define here and provide meaning.
Page 9. Instead “concrete form” it’s better to use special or particular.
Page 9. Formula (13). Please define x0 here.
Page 10. Are tau0 and size x0, defined as the age and size at the entry into puberty, correspond to the age and size at 50% maturity that are more widely used in fisheries literature? The authors allude to this interpretation on page 15. By the way, it seems that tau0 is defined by not used anywhere.
Page 10. Should equation (15) be written as an ordinary differential equation since in the equation for a stationary state of model (14) the variable y depends only on x?
Page 10. The expression for growth is already given by (12) and for natural mortality function is already given by (13), no need to repeat them here.
Page 10. Is it really necessary to consider the whole interval [0,1) for the proof of limit property at x->1, given also that the integral over [0,x0] is dropped into a constant anyway? Considering the mortality term for the adult part of the population only, the equilibrium solution can be written in a more concise form, allowing the proof of the proposition.
Page 10. Please state explicitly why the functional form of m1(x) has changed. Supposedly it has something to do with the Heaviside function that becomes zero for sizes above x0, but this should be stated.
Page 10, at the very bottom. The constant a3 appears here first, so put it together with this formula for y(x).
Page 11. Please provide some connection statements to clarify what was done. This is a broader comment related to overall math in the text. For example, here it can be written “which simplifies to ”.
Page 11. “to form equation”? Remove ‘form’. Also, no need to repeat equation (15) below.
Page 11. Please add the number to the formula for the birth rate for clarity as it is cited in couple of places below.
Page 12. “scenario is based on this condition” What condition?
Page 13. Where y2(t) was defined? Should it be y1(t)?
Page 13. “The diffusion coefficient…” So, was it actually non-zero in numerical experiments?
Page 13. The seasonality function used in eq. for y1(t) also controls the catch, eq. (19), and hence the profit (20). So, the fishing occurs synchronously with spawning? Is it justified by real data or it’s a model assumption?
Page 13. The function q(x) seems to refer to catchability coefficient q0 multiplied by gear selectivity (modeled here with with Heaviside function). Perhaps, worth adding a little interpretation here.
Page 13. What is p0 in the income from the fishery function?
Page 13. Cannot be done easily.
Page 14. At the beginning of the second paragraph, add an introductory sentence for clarity, e.g., “Let us now consider the model with harvesting.”
Page 14. Same remark about ‘high gradients’ for u(t).
Page 14. The sentence about constant environment is confusing. Aren’t these results supposed to be obtained for the model with variable environmental conditions impacting recruitment? Please revise this paragraph to improve clarity.
Page 15. “This condition indicates… ” In the context it is not condition, but requirement.
Page 17. “The abundance fluctuations”. The population abundance (as the integral of density over x), or the density, y(x), fluctuations?
Page 17. The first paragraph provides some key modeling outcomes. First two are pretty obvious and demonstrated by the results, but the third one, although also can be derived from Figure 7, would benefit from some quantitative evaluations, that is calculating the mean size in the scenario without and with fishing. Also, the last results about density fluctuations. Are lower amplitudes explained by the decrease of the stock size?
Page 17. “An unstable environment has minimal influence”. Please clarify.
Author Response
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Author Response File: 
Author Response.docx
Reviewer 2 Report
Being a fishery scientist, I must confess that I do not fully understand the mathematics in this manuscript. However, I have a general impression that this is a good article, in Figures 2-7, there are all two peaks which may indicate that the models and data agree with each other well. From the fisheries point of view, I would recommend to e.g. include a case study of a pelagic fish species, as well as to discuss the pros and cons of the classical fish spawner-recruit models of such as Ricker model and Beverton-Holt model.
Author Response
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Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Thanks to the authors for thorough revision, including multiple clarifications, discussion and better structuring of the manuscript. Thanks to this revision, the description of this very interesting and comprehensive study has been significantly improved. I think this paper will be of great interest to the fisheries scientists and mathematical modelers. I therefore recommend accepting the manuscript after a very minor revision (see below), which however, does not require further review.
In response to the answers provided by the authors, here are some feedback and suggestions:
Authors: Various fishery strategies have been developed in fishery theory. Is it better?
Reviewer: Not really, the specific problem is not highlighted and the wide scope of references do not add clarity either. In fact the whole paragraph could benefit from some revision to make it more specific and focused. To illustrate the claim about specificity, here's how it could be potentially rewritten:
The impact of harvesting on populations is a fundamental problem in fishery theory. How harvesting affects population dynamics depends not only on stock structure and biological parameters of population, but also on fishing strategies. The over-exploitation, occurring as a result of multiple factors, undermines the conservation of populations and also poses the problem for fishery. That is why the optimal harvesting problem should be considered within the models describing population dynamics and fishing.
Authors: Yes, it is ordinary differential equation. Equation (15) describes a stationary case (equilibrium).
Reviewer: The point was exactly related to the use of notation of partial derivative for the ordinary derivative.
Authors: The parameter p0 is the highest specific income from harvesting individuals of size x0. As the size increases, the income decreases.
Reviewer: It would be better to have this explanation in the text.
Authors: Yes, here we speak about density.
Reviewer: Then to avoid ambiguity, it would be better to say 'density fluctuations' in the text.
Author Response
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Author Response File: Author Response.docx
