A MODEL FOR FISHERY RESOURCE WITH RESERVE AREA AND FACING PREY-PREDATOR INTERACTIONS

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 14, Number 4, Winter 2006 A MODEL FOR FISHERY RESOURCE WITH RESERVE AREA AND FACING PREY-PREDATOR INTERACTIONS T. K. KAR ABSTRACT. In this paper, we propose and analyse a nonlinear mathematical model to study the dynamics of a fishery resource system in an aquatic environment that consists of two zones: a free fishing zone and a reserve zone where fishing is strictly prohibited. Biological equilibria of the system are obtained, and criteria for local stability and global stability of the system are derived. An optimal harvesting policy is also discussed using the Pontryagin s Maximal Principle. 1 Introduction The excessive and unsustainable exploitation of our marine resources has led to the promotion of marine reserves as a fisheries management tool. Marine reserves, areas in which fishing is restricted or prohibited, can offer opportunities for the recovery of exploited stock and fishery enhancement. Various achievements are expected from the creation of marine reserves see [24], [36]). The objectives pursued can generally be classified under one of the following three categories: ecosystem preservation, the management of commercial fisheries, and/or the development of recreational activities. At such a general level, the degree of compatibility between the three objectives is difficult to assess. It is bound to vary from case to case, depending on the exact conditions under which the plan to create a marine reserve is being discussed. In some cases, the pursuit of these objectives can lead to conflicts between the various economic agents involved in the creation of a protected area see [15]). This makes it important to develop tools which allow the overall assessment of the consequences of creating a marine reserve, taking into account the distribution of impacts amongst the various categories of activities and economic agents involved. Keywords: Fishery resource, marine reserve, global stability, optimal harvesting policy. Copyright c Applied Mathematics Institute, University of Alberta. 385

2 386 T. K. KAR The fast growing literature on marine reserves has largely centred on the issue of commercial fisheries management see [7, 8], [21], [34]). Bohnsack [6] gives a comprehensive list of the potential biological effects that may be anticipated from the establishment of a marine reserve, which includes: i) protection of spawning biomass, ii) providing a recruitment source for surrounding areas, iii) supplemental restocking of fished areas through emigration, iv) maintenance of natural population age structure, v) maintenance of areas of undisturbed habitat, and vi) insurance against management failures in fished areas. These biological effects are expected to make fish stocks more resilient to exploitation, thereby reducing the risk of stock collapse [9], [17], [22], [35]. Marine reserves may in addition be regarded as a means of dealing with uncertainties related to stock assessment and effort control in fisheries management [12, 29]. A number of studies have focused on the development of biological models see [14, 31, 37]) and bio-economic models of marine reserve see [1, 2, 13, 23, 30]). In a renewable resource e.g., fishery and forestry) management, maximizing the present value of benefits derived from the resources and its conservation are important problems to be studied. During the last decade several investigations regarding fishery resource have been conducted [4, 20, 10, 33, 18]. Recently, Kar and Chaudhuri [26] presented a mathematical model of nonselective harvesting in a prey-predator fishery. In their further work [27] they described the regulation of a prey-predator fishery by taxation as a control instrument. Kar [28] also presented a model to study the selective harvesting in a prey-predator fishery by introducing a time delay in the harvesting term. Dubey, et al. [16] and Fan and Wang [19] proposed harvested population with diffusional migration. Commonly, when predators are placed together with their prey species in a homogenous space, their dynamics bring the prey density to a very low value [3]. As a result, the prey becomes extinct, followed closely by the predator. When a predator species causes the extinction of its prey in a homogeneous environment, the predation must be very intense. A refuge patch, where the preys are free from predation, will allow the co-existence of the two species. Marine reserves, areas in which fishing is restricted or prohibited, can offer opportunities for the recovery of exploited stock and fishery enhancement. In this paper, we model this phenomenon in an aquatic

3 A MODEL FOR FISHERY RESOURCE 387 habitat that consists of two zones: one free fishing zone and the other a reserve zone where fishing is not permitted. The choice of this model was motivated by the existence of Marine National Park, Kenya, a fully protected coral reef marine reserve comprising approximately 30% of former fishing ground and Marine National Park in the Iroise Sea, a costal sea west of Brittany France). We derive conditions for the existence of steady states and study their stability behaviour. Finally, we discuss the optimal harvesting policy using Pontraygin s Maximum Principle. 2 Description of the model We study a prey-predator system in a two patch environment: one accessible to both prey and predators patch 1) and the other one being a refuge for the prey patch 2). Each patch is supposed to be homogenous. The prey refuge patch 2) constitutes a reserve area of prey and no fishing is permitted in the reserved zone while the unreserved zone area is an open-access fishery zone. We suppose that the prey migrate between the two patches randomly. The growth of prey in each patch in the absence of predators is assumed to be logistic. Keeping these, in view, model becomes dx 1 dt = rx x ) σ 1 x + σ 2 y mxz K a + x qex, dy 1 dt = sy y ) 2.1) + σ 1 x σ 2 y, L dz dt = dz + mαxz a + x. Here, xt) and yt) are the biomass densities of the prey species inside the unreserved and reserved areas, respectively, at time t. zt) is the biomass density of predator species at time t. Migration rate from unreserved area to reserved area and reserved area to unreserved area are σ 1 and σ 2, respectively. The effort applied for harvesting the fish population in the unreserved area is E. The intrinsic growth rates of prey species inside the unreserved and reserved patches are r and s, respectively. The carrying capacities of prey species in the unreserved and reserved areas are K and L, respectively. The maximal relative increase in predation is m; a, the Michaelins-Menten constant, is the conversion factor we assume α < 1, since the whole biomass of the prey is not transformed to the biomass of the predator). Traditionally, the predation term is taken as mxy. Here mx may be interpreted, in a wider sense, to be the trophic, or the functional, response of the predator to the density of prey. This response function

4 388 T. K. KAR is termed as Holling Type I response function. We do not take the predation term in this form because it embodies the unrealistic feature that predation tends to infinity as the prey population goes to infinity with finite and fixed predator population. Therefore, we have taken the predation term as mxy/a + x) due to Holling [25]. The response function mx/a + x) is termed as Holling Type II response function. Here we point out that lim x mxy a + x = my. 3 Existence of equilibria Equilibria of model 2.1) is obtained by solving ẋ = ẏ = ż = 0. This gives three possible steady states, namely, P 0 0, 0, 0), P 1 x, y, 0) and P 2 x, y, z ). Here x and y are the positive solutions of 3.1) Eliminating y we get rx 1 x ) σ 1 x + σ 2 y mxz K a + x sy 1 y L 3.2) a 1 x 3 + b 1 x 2 + c 1 x + d 1 = 0 where a 1 = sr2 Lσ 2 2 K2, b 1 = 2sr r σ 1 qe) Lσ 2 2 K, c 1 = s r σ 1 qe) 2 Lσ 2 2 qex = 0, ) + σ 1 x σ 2 y = 0. s σ 2) r Lσ 2, d 1 = s σ 2) σ 2 r σ 1 qe) σ 1. Equation 3.2) has a unique positive solution x = x if the following inequalities hold: 3.3) s r σ 1 qe) 2 Lσ 2 < s σ 2) r, K s σ 2 ) r σ 1 qe) < σ 1 σ 2.

5 A MODEL FOR FISHERY RESOURCE 389 Substituting x, we get y from 3.1) and to be positive, we must have 3.4) x > K r r σ 1 qe). Again, x, y and z are positive solution of 3.5a) 3.5b) 3.5c) rx 1 x ) σ 1 x + σ 2 y mxz K a + x sy 1 y L qex = 0, ) + σ 1 x σ 2 y = 0, d + mαx a + x = 0. From 3.5c) we get x = ad/α d), which is positive if mα > d. Substituting the value of x in 3.5b) we get s L y2 s σ 2 ) y σ 1ad mα d = 0. The above equation has a unique positive solution y = y if s > σ 2. Knowing the value of x* and y*, z* can be obtained from 3.5a) as ) ) } a + x Z = {rx 1 mx x σ 1 x + σ 2 y qex. K It may be noted that for z to be positive, we must have y > rx 2 K rx + σ 1 x + qex. 4 Dynamical behaviour of equilibria The dynamical behaviour of equilibria can be studied by computing variational matrices corresponding to each equilibrium. We note that the trivial equilibrium P o is unstable. P 1 x, y, 0) is locally asymptotically stable if d > mαx/a+x). The characteristic equation about P 2 x, y, z ) is 4.1) λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0 where a 1 = rx K + σ 2y x + σ 1x y + sy L + d mz x a + x ) 2 mαx a + x,

6 390 T. K. KAR a 2 = a 3 = rx K + σ 2y x mz x a + x ) 2 σ1 x + y + sy L rx K + σ 2y x mz x + m2 aαx z a + x ) 3 ) σ1 x ) y + sy L + d mαx a + x ) )d mαx a + x + m2 aαx z a + x ) 3 σ 1σ 2, ) σ1 x ) a + x ) 2 y + )d sy mαx L a + x σ1 x ) ) y + sy σ 1 σ 2 d mαx L a + x. By the Routh-Hurwitz criterion, it follows that all eigenvalues of 4.1) have negative real parts if and only if a 1 > 0, a 3 > 0 and a 1 a 2 > a 3. In the following lemma we show that all solutions of model 2.1) are positive and uniformly bounded. Lemma 4.1. The set Ω = { x, y, z) R 3 + : w = x + y + 1/α)z µ/v} is a region of attraction for all solutions initiating in the interior of the positive octent, where v > d is a positive constant and µ = K 2r r + v qe)2 + L 2s s + v)2. Proof. Let wt) = xt) + yt) + 1/α)zt) and v > 0 be a constant. Then we have dw dt + vw = r K x2 + r + v qe)x s d L y2 + s + v)y α v ) z α K 2r r + v qe)2 + L s + v)2 2s = µ, say). By the theory of differential inequality by Birkoff and Rota [5], we have 0 < w xt), yt), zt)) µ v 1 e vt ) + w x0), y0), z0)) e vt, and when t, 0 < w µ/v, proving the lemma.

7 A MODEL FOR FISHERY RESOURCE 391 In the following theorems we discuss the global stability of equilibria P 1 x, y, 0) and P 2 x, y, z ). Theorem 4.1. The equilibrium point P1 is globally asymptotically stable. Proof. Let us consider the following Lyapunov function V = x x x ln x ) + K 1 y y y ln y ). x y Differentiating V with respect to time t, we get dv dt = x x dx x dt + K y y dy 1 y dt. Let us choose K 1 = y/x)σ 2 /σ 1 ). Then, dv dt = r K x x)2 y xσ 1 L y y)2 σ 2 xxy xy xy)2 < 0. Therefore, P 1 x, y, 0) is globally asymptotically stable. The above theorem implies that in the absence of predators in an open-access fishery region, if a subregion is reserved where fishing is not allowed and fish populations are harvested only outside the reserved subregion, then in both the reserved and unreserved zones fish species settle down to their respective equilibrium levels. This implies that fish populations may be sustained at an appropriate equilibrium level even after continuous harvesting of fish population in unreserved area. Theorem 4.2. The positive equilibrium P 2 x, y, z ) is globally asymptotically stable if σ1 x αaσ2 y ) 2 rs < 2 KL αa. Proof. We make use the standard Lyapunov function V = V x, y, z) = L 1 x x x ln x x ) + L 2 y y y ln y y )

8 392 T. K. KAR + L 3 z z z ln z z ), where L 1, L 2 and L 3 are positive constants to be determined in the subsequent steps. Differentiating V with respect to t we get where dv dt = L x x dx 1 x dt + L y y dy 2 y dt + L z z dz 3 z dt = r K L 1x x ) 2 s L 2 L y y ) 2 L 1 m a + x)a + x ) x x )z z ) + L 3 mαa a + x)a + x ) x x )z z ) Γx, y), Γx, y) = L 1yσ 2 xx x x ) 2 + L 2xσ 1 yy y y ) 2 L1 σ 2 x + L ) 2σ 1 y x x )y y ). If we choose L 3 = 1, L 1 = αa, L 2 = 1, then we have dv dt = r K L 1x x ) 2 s L L 2y y ) 2 Γx, y). It is easy to show that Γx, y) Thus we have, L 1 L 2 σ 1 σ 2 2 x y σ 1 x σ ) 2 y x x )y y ) L2 σ 1 = x ) 2 L 1 σ 2 y x x )y y ). dv dt r K L 1x x ) 2 s L L 2y y ) 2 L2 σ 1 x ) 2 L 1 σ 2 y x x )y y )

9 A MODEL FOR FISHERY RESOURCE 393 Clearly, if L2 σ 1 x ) 2 L 1 σ 2 rs y < 2 KL L 1L 2, is globally asymptot- then for x, y, z) x, y, z ), dv dt. Therefore, P 2 ically stable. 5 Optimal harvesting policy Our objective is to maximize the present value J of a continuous-time stream of revenues given by 5.1) J = 0 e δt pqxt) c) Et) dt where δ is the instantaneous rate of annual discount. Thus, our objective is to maximize J subject to state equations 2.1) and to the control constraints 5.2) 0 Et) E max. To solve this optimization problem, we utilize the Pontryagin s Maximal Principle [32]. The associated Hamiltonian is given by [ 5.3) H = e δt pqx c)e + λ 1 t) rx 1 x ) σ 1 x K + σ 2 y mxz ] a + x qex + λ 2 t) [ sy 1 y L ) ] + σ 1 x σ 2 y + λ 3 t) where λ 1, λ 2 and λ 3 are adjoint variables and σt) = e δt pqx c) λ 1 qx [ dz + mαxz ], a + x is called the switching function [11]. Since H is linear in the control variable E, the optimal control will be a combination of bang-bang control and singular control. The optimal control Et) which maximizes H must satisfy the following conditions: 5.4a) E = E max, when σt) > 0, i.e., when λ 1 e δt < p c qx,

10 394 T. K. KAR 5.4b) E = 0, when σt) < 0, i.e., when λ 1 e δt > p c qx. λ 1 e δt is the usual shadow price [11] and p c/qx)is the net economic revenue on a unit harvest. This shows that E = E max or zero according to the shadow price is less than or greater than the net economic revenue on a unit harvest. Economically condition 5.4a) implies that if the profit after paying all the expenses is positive, then it is beneficial to harvest up to the limit of available effort. Condition 5.4b) implies that when the shadow price exceeds the fisherman s net economic revenue on a unit harvest, then the fisherman will not exert any effort. When σt) = 0, i.e., when the shadow price equals the net economic revenue on a unit harvest, then the Hamiltonian H becomes independent of the control variable Et), i.e., H/ E = 0. This is the necessary condition for the singular control E t) to be optimal over the control set 0 < E < E max. Thus, the optimal harvesting policy is E max, σt) > 0 5.5) Et) = 0, σt) < 0 E, σt) = 0 when σt) = 0. It follows that 5.6) λ 1 qx = e δt δt Π pqx c) = e E. This implies that the user s cost of harvest per unit of effort equals the discounted value of the future marginal profit of the effort at the steady-state level. Now, in order to find the path of singular control we utilize the Pontryagin s Maximum Principle, and the adjoint variables λ 1, λ 2 and λ 3 must satisfy 5.7) 5.8) dλ 1 dt dλ 2 dt = H x = [e δt pqe + λ 1 { r 2rx K σ 1 = H y = mαaz λ 2 σ 1 λ 3 a + x) 2, [λ 1 σ 2 + λ 2 s 2sy L σ 2 maz }] a + x) 2 qe )],

11 A MODEL FOR FISHERY RESOURCE ) dλ 3 dt = H [ z = λ 1 mx a + x + λ 3 d + mαx )]. a + x Considering the interior equilibrium P 2 x, y, z ) and Equation 5.6), Equation 5.8) can be written as where whose solution is given by dλ 2 dt A 1λ 2 = A 2 e δt A 1 = sy L + σ 1x y, A 2 = σ 2 p c ) qx 5.10) λ 2 = A 2 A 1 + δ e δt. From 5.9) we get 5.11) λ 3 = mx a + x)δ From 5.7) we get where whose solution is given by p c ) e δt. qx dλ 1 dt B 1λ 1 = B 2 e δt B 1 = rx K + σ 2y x mzx a + x) 2, B 2 = pqe + A 2σ 1 A 1 + δ m2 aαxz a + x) 3 δ 5.12) λ 1 = B 2 B 1 + δ e δt. From 5.6) and 5.12) we get the singular path 5.13) p c ) = B 2 qx B 1 + δ. p c ), qx Equation 3.5) together with equation 5.13) gives the optimal equilibrium population x = x δ, y = y δ and z = z δ. Then the optimal harvesting effort is given by E = E δ = 1 q [ r 1 x δ K ) σ 1 + σ 2 y δ x δ mz δ α + x δ ].

12 396 T. K. KAR Remark 1. From Eqns. 5.10), 5.11) and 5.12), we note that λ i t)e δt, i = 1, 2, 3) is independent of time in an optimum equilibrium. Hence they satisfy the transversality conditions at, i.e., they remain bounded as t. Remark 2. From 5.6), we note that δt Π λ 1 qx = e E where Π/ E is evaluated at the positive equilibrium level. Π/ E, the rate of change of economic revenue, is called marginal revenue of the effort. The product of the discounting factor and the rate of change of economic revenue i.e., the right hand side) gives the present value of the marginal revenue of effort and left hand side is the total user s cost of harvest per unit of effort. Thus we conclude that the optimum euqilibrium solutions is obtained when total user s cost equals the present value of marginal revenue of effort. Remark 3. From 5.13) we also have pqx c = B 2qx 0 as δ. B 1 + δ Thus, the net economic revenue pqx c = 0. This implies an infinite discount rate leads to the net economic revenue to zero and hence the fishery would remain closed. Simulation For simulation let us take r = 3.0, K = 110, σ 1 = 0.5, σ 2 = 0.5, m = 2.5, a = 12.0, q = 0.01, s = 0.4, L = 200, d = 0.01, α = 0.006, p = 15, c = 1.4, δ = For the above values of parameters we find the optimal equilibrium as 24.0, 56.4, 16.2). Using the above parameters, the sensitivity analysis is performed to study the effect of predation on optimal solution. m x δ y δ z δ Thus we observed that, as the predation increases optimal equilibrium levels decreases for a given effort.

13 A MODEL FOR FISHERY RESOURCE Concluding remarks In this paper, a mathematical model has been proposed and analyzed to study the dynamics of a prey-predator fishery model with prey dispersal in a two patch environment: one is assumed to be free fishing zone and the other is the reserved zone. It has also been assumed that predation takes place in the unreserved zone only. We have discussed the local and global stability of the system. It has been observed that in the absence of predator, even under continuous harvesting in the unreserved zone, the fish population may be maintained at an appropriate equilibrium level. Using the Pontryagin s Maximum Principle, the optimal harvesting policy has been discussed. It has been found that the total user s cost of harvest per unit of effort equals to the discounted value of the future marginal profit of the effort at the steady-state level. It has also been noted that if the discounted rate increases, then the economic rent decreases and even may tend to zero if the discounted rate tend to infinity. The possibility that the predator species in this model be itself harvested for commercial purposes, or that its non-use value be an important element in the overall economic value of the marine ecosystem have not been considered in this paper. Their inclusion in the model would surely lead to characterise another set of trade-offs which are bound to be at the centre of marine reserve policies. Acknowledgement The author would like to thank Japan Society for the Promotion in Sciences JSPS) for financially supporting the research P05109) and Professor Hiroyuki Matsuda, Yokohama National University, for which all that have been possible. The author would also like to thank the anonymous referee for giving some constructive comments to improve the contents of the paper. REFERENCES 1. L. G. Anderson, Marine reserves: a closer look at what they can accomplish, 10th Biennal Conference of IIFET, July 10 14, Corvallis, Oregon, USA, R. Arnason, Marine reserves is there an economic justification?, Conference on the economic of marine protected areas at the UBC Fisheries Center, Vancouver, July 6 7, A. A. Berryman, The origin and evolution of predator-prey theory, Ecology ),

14 398 T. K. KAR 4. D. K. Bhattacharya and S. Begum, Bionomic equilibrium of two species system, Math. Biosci. 1352) 1996), G. Birkoff and G. C. Rota, Ordinary Differential Equations, Ginn, J. A. Bohnsack, The potential of marine fishery reserves for reef fish management in the U.S. Southern Atlantic, NOAA Tech. Memo. NMFS-SEFC-261, 40 p., J. A. Bohnsack, Marine reserves: They enhance fisheries, reduce conflicts and protect resources, Oceanus 363) 1993), J. A. Bohnsack, Marine reserves, zoning, and the future of fishery management, Fisheries 219) 1996), M. H. Carr and D. C. Reed, Conceptual issues relevant to marine harvest refuges: examples from temperate reef fishes, Canadian Journal of Fisheries and Aquatic Sciences ), J. Chattopadhyay, A. Mukhopadhyay and P. K. Tapasai, A resource based competitive system in three species fishery, Nonlinear Studies ), C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd ed., John Wiley, New York, C. W. Clark, Marine reserves and the precautionary management of fisheries, Ecological Applications 62) 1996), J. M. Conrad, The bioeconomics of marine sanctuaries, Journal of Bioeconomics ), E. E. DeMartini, Modelling the potential of fishery reserves for managing pacific coral reef fishes, Fishery Bulletin ), J. A. Dixon, Economic benefits of marine protected areas, Oceanus 363) 1993), B. Dubey, P. Chandra and P. Sinha, A resource dependent fishery model with optimal harvesting policy, J. Biol. Syst ), J. E. Dugan and G. E. Davis, Application of marine refugia to coastal fisheries management, Canadian Journal of Fisheries and Aquatic Sciences ), M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci ), M. Fan and K. Wang, Study on harvested population with diffusional migration, J. Syst. Sci. Comp. 142) 2001), S. Ganguli and K. S. Chaudhuri, Regulation of a single-species fishery by taxation, Ecol. Model ), S. Gubbay, Marine refuges: the next step for nature conservation and fisheries management in the North-East Atlantic, A report for WWF-UK, 1996,22 p. 22. S. Guenette, T. Lauck and C. W. Clark, Marine reserves: from Beverton and Holt to the present, Reviews in Fish Biology and Fisheries ), R. Hannesson, Marine reserves: what would they accomplish?, Marine Resource Economics ), P. Hoagland, Y. Kaoru and J. M. Broadus, A methodological review of net benefit evaluation for marine reserves, The World Bank, Environment Department, ). 25. C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can ), T. K. Kar and K. S. Chaudhuri, On non-selective harvesting of a multispecies fishery, Int. J. Math. Educ. Sci. Technol. 334) 2002), T. K. Kar and K. S. Chaudhuri, Regulation of a prey-predator fishery by taxation: a dynamic reaction model, J. Biol. Syst ), T. K. Kar, Selective harvesting in a prey-predator fishery with time delay, Math. Comp. Model ),

15 A MODEL FOR FISHERY RESOURCE T. Lauck, C. W. Clark, M. Mangel and G. R. Munro, Implementing the precautionary principle in fisheries management through marine reserves, Ecol. Appl ), Suppl. 1): S72 S J. Pezzey, C. Roberts and B. Urdal, A simple bioeconomic model of a marine reserve, Ecological Economics ), T. Pollacheck, Year around closed areas as a management tool, Natural Resource Modeling 43) 1990), L. S. Pontryagin, V. G. Boltyonskii, R. V. Gamkrelidre and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, T. Pradhan and K. S. Chaudhuri, A dynamic reaction model of a two species fishery with taxation as a control instrument: a capital theoretic analysis, Ecol. Model ), C. M. Roberts and N. V. C. Polunin, Marine reserves: simple solutions to managing complex fisheries, Ambio 226) 1993), R. J. Rowley, Case studies and reviews: Marine reserves in fisheries management, Aquatic Conservation: marine and freshwater ecosystems ), N. Shackell and M. J. H. Willison, eds. Marine protected areas and sustainable fisheries, Science and Management of Protected Areas Association, Wolfville, Nova Scotia, Canada, J. Sladek Nowlis and C. M. Roberts, Fisheries benefits and optimal design of marine reserve, Fishery Bulletin 973) 1999), Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , INDIA address: tkar1117@gmail.com; t k kar@yahoo.com Present address From 11th May th May 2007) Faculty of Environment and Information Sciences, Yokohama National University, 79-7, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa , JAPAN address: tkar1117@gmail.com

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