Optimal pricing strategy for livestock of fishery and poultry

Economic Modelling 29 (2012) 1024–1034

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Optimal pricing strategy for livestock of fishery and poultry Shib Sankar Sana ⁎ Department of Mathematics, Bhangar Mahavidyalaya, University of Calcutta, Bhangar,Pin-743502, 24PGS(South), West Bengal, India

a r t i c l e

i n f o

Article history: Accepted 7 March 2012 Keywords: Control theory Dynamical system Sales price Equilibrium Deterioration Stability

a b s t r a c t The paper deals with a joint project of fishery and poultry while growth rates of both the species depend on the available nutrients and environmental carrying capacities of biomasses. The demand rates of both the species in the market vary with the selling prices and on-hand stock of the species. The existence of steady states and its stability (local and global) analysis are studied in details. The relevant profit of the project is maximized with the help of Pontryagin's Maximum Principle. The model is justified by a suitable numerical example. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The integrated project of fisheries and poultries has received great attention from researchers and practitioners of firm management. The formulation of a realistic model of multi-species community is a growing field of research nowadays. Clark (1985) has discussed the model of combined non-selective harvesting of two ecologically independent populations, using the logistic law of growth. Ragozin and Brown (1985) have investigated the optimal policy for harvesting the predator of predator–prey system while the prey has no market value by virtue of being uncatchable. Mesterton-Gibbons (1988) has extended the work of Ragozin and Brown (1985) by discussing the optimal policy for combined harvesting of both predator and prey. Mesterton-Gibbons (1996) has described a technique to obtain the optimal harvesting policy for a Lotka–Volterra ecosystem of two interdependent populations when harvesting rate is proportional to the harvest effort and either a single stock is selectively or both stocks are harvested together. Mesterton-Gibbons (1988) and MestertonGibbons (1996) have solved for the singular control and described the approach to equilibrium for a generalized ecological interaction. Hoagland and Jin (1997) partially solved the same generalized interacting-species case while on species have extractive value and the other existence value. Hoagland et al. (2003) have examined the interactions between marine (or estuarine) aquaculture and a wild harvest fishery. In this connection, they have developed a framework to analyze the tradeoffs between a wild harvest fishery and aquaculture occurring in the same region and selling into the same market. Jin et al. (2003) have developed an economic–ecological model by

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merging an input–output model of coastal economy with a model of a marine food web. According to them, specific environmental and natural resource parameters may be calculated by the ecological model and then used as input parameters for the economic model. Herrera (2006) has analyzed the nonselective harvest of two stocks with generalized ecological interaction and different persistent distributions across two spatial strata. In his model, an aggregate effort control, depending on harvest response was shown to partially dissipate rents relative to the case where the spatial distribution of effort could be specified. The harvesting of population species is commonly practiced in fisheries, forestry, and wildlife management of renewable resources. Problems related to the exploitation of multi-species systems are not only interesting but also difficult as there are theoretical as well as practical difficulties in the determination of an optimal policy for the harvesting of multi-species system. In a competitive market, the quality of a product and fair price in any business organization play an important role in marketing management. The good (conforming) quality of the product is sold in the market to keep brand image of the enterprise, whereas the non-conforming quality (deteriorated) items are used in another purpose at less marketing value. Generally speaking, deterioration is considered as the result of various effects on stock such as damage, decreasing usefulness in original purpose and many more. Goyal and Gunasekaran (1995) have developed an integrated production– inventory–marketing model for determining the EPQ (economic production quantity) and EOQ (economic order quantity) for raw materials in a multi-stage production system. This model considered the effect of different marketing policies such as the price per unit product and the advertisement frequency on the demand of a perishable item. Recently, Sana (2010a) has formulated an EOQ model over an infinite time horizon for deteriorating items while the demand is price-sensitive, considering partial backordering and time dependent

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

deterioration rate. Sana (2010b) has established a multi-item EOQ model both for ameliorating and deteriorating items with capacity constraint for storage, assuming the rate of demand is a function of enterprises' initiatives. Other papers related to this area are CardenasBarron (2000), Goyal and Cardenas-Barron (2002), Khanra and Chaudhuri (2003), Sana and Chaudhuri (2004), Mukhopadhyay et al. (2004), Mukhopadhyay et al. (2005), and Ghosh and Chaudhuri (2006), among others. The pricing of sales commodities is a focal point in any business organization in a given economy. Abad (1996) has established the optimal pricing and lot-sizing EOQ policies under conditions of perishability and partial backordering. Abad (2000) has extended the optimal pricing and lot-sizing EOQ model to an economic production quantity model. Arcelus et al. (2003) have generalized the special-sales problem, by assuming temporary discounts. They have also incorporated other practical concerns, most notably a pricedependent demand, in tune also to the current tenets of microeconomic theory. Papachristos and Skouri (2003) have extended the EOQ model for selling price sensitive demand with quantity discount, pricing and partial backordering. Teng and Chang (2005) have developed an EPQ (Economic Production Quantity) model for deteriorating items when the demand rate depends on stock-display and selling price. It is quite natural that the poultry litter may be used as feed of fishery and the small fishes which have less market value are used as nutrient of poultry birds. The poultry litter is excreta, feathers, spilled feed, soil and dead birds. The U.S. produces approximately 8.7 billion broiler chickens annually that results in 13–26 million metric tons of poultry litter (i.e., excreta, feathers, spilled feed, bedding material, soil and dead birds) (Moore et al., 1995). It is estimated that nearly 80% of poultry units in the U.S. use antibiotics in feed (Silbergeld et al., 2008). The poultry litter contains large amounts of antibiotic resistant bacteria and resistance genes associated with the use of antibiotics in poultry production (Nandi et al., 2004). This has raised the concern of environmental dispersal of antibiotic resistance. Poultry litter is generally piled between 1 and 4 m deep and stored in open sheds until it is used in land as a soil amendment. Population of house flies grows in the poultry litter and those participate in the dispersion of antibiotic resistance from poultry houses into the environment (Winpisinger et al., 2005). Synanthropic flies have evolved to live in proximity to humans and have been found to carry a number of different pathogenic microorganism, including viruses and bacteria, and can play a major role in the epidemiology of infections in humans body (Likirdopulos et al., 2005; Macovei and Zurek, 2006; Nichols, 2005). Flies may also play an important role in spreading avian influenza. In Japan, researchers have reported that flies were captured in proximity to broiler facilities during an outbreak of highly pathogenic avian influenza in Kyoto in 2004. H5N1 influenza viruses are found in the chickens of the infected poultry farm which are carried by flies in poultry litter (Sawabe et al., 2004). Graham et al. (2008) have investigated that large scale broiler poultry production results in many obstacles to biocontainment. Antibiotic resistant Enterococci and Staphylococci have been isolated from poultry litter (Hayes et al., 2004; Lu et al., 2003; Simjee et al., 2007). Furthermore, owing to methods of waste storage at farms, large amount of fresh and stored poultry litter available outside the poultry houses can serve also as a substrate for development of fly populations. Therefore, the fresh poultry litter may be used as nutrients of fishes that are economically beneficial to the farms as well as the environmental protection. The integration of poultry and fishery can increase overall production intensity and economies on land, labor and water requirements for both poultry and fish. For example, the wastes of 1500 poultry can produce 10 metric ton fish in 1 ha of static water fish ponds without other feeds or fertilizers (FAO, 2003). Poultry waste/litter contains more nutrients than other livestock wastes. It contains less moisture, fiber and compounds such as fish pond fertilizers. These nutrients can be used as fertilizer of the pond that stimulates production of natural food organisms such as phytoplankton and detritus. A variety of carps, shrimp and tilapias (Oreochroms Sp.) can grow rapidly on such natural feeds alone. Poultry litter can be used

1025

fresh, or after processing, to enhance natural food production in sun-lit tropical ponds. Stable and high water temperature and sunlight ensure year-round growth of fish and their natural feeds. The tropics where average temperatures remain above 25 °C are ideal for culturing fish using poultry waste as nutrient, although it is also practiced in subtropical and sub-temperature (>20 °C) climates during suitable periods of the year. Poultry processing byproducts such as chicken bones, intestines and whole carcasses have greater value as direct feeds and are normally used for higher value fish species. Most of the poultry and fishery systems, poultry litter raised on balanced feeds give the most nutrient-rich waste and produce the fish, but systems are frequently suboptimal resulting in inefficient waste or space use. Poultry manure is used either directly on-site, through the siting of poultry houses over ponds, or after collection, storage and transport to the site of fish culture. Construction of the poultry house over the pond allows to drop directly in, saving labor costs. In the peri-urban, flood-prone land is often used in which the cost to fill land for poultry housing and the opportunity cost of land itself are reduced. Singholka (1979) and Michael (1980) have shown that prawns feed chicken feed, or ground fish flesh mixed with cooked broken rice, beef, hog, etc. The supplemental food (broken rice, dead poultry, beef, hog, fish processing waste, prawn processing wastes, snails, etc.) acts as a fertilizer and increases the biological productivity of the pond as a whole rather than acting as a true prawn feed. The pelleted food added is eaten by small fish in microbrachium culture which themselves form a source of food for the prawns. The poultry litter is nutrient-rich, but there is a great variability in quality at the time of use as fish production inputs (Taiganides, 1979). Fish species play an important role in determining loading rates of poultry waste because of its value for fertilization of the pond. The nutrient value of the waste is measured by the rate of nitrogen (dissolved inorganic nitrogen), phosphorous (soluble reactive phosphorous) and dissolved oxygen release. The production of excreta by poultry varies widely live weight and productivity, diet and water intake, as well as housing and seasonal weather conditions, are all factors which can influence the total quantity and nutrient content of excreta product by livestock. Air-breathing fish, such as clarias catfish, silver-striped catfish and Pangasius hypothalmus can tolerate highest input levels and they also require extra feed to sustain growth. These fish species are also probably inefficient to take phytoplankton dominated food web. Waste products from fish-processing operations and the underutilized species of fish to produce fish protein for animal feed are common phenomena in poultry farming. Hassan and Heath (1986) have given information to facilitative production of fish silage from waste and under-utilized fish by biological fermentation for potential use on animal and poultry feed. Hassan et al. (1997) have experimented the suitability of poultry-feather meal as a substitute for dietary fish meal protein in the diet of Indian major carp. They show that hydrolyzed poultry-feather meal can be used at 20% of dietary level (i.e., 50% protein level) for Labio rohita fry without compromising growth and feed utilization. Effects of inflation and time value of money can no longer be ignored in the present economy because large scale inflation and time value of money decline the purchasing power of money sharply. Buzacott (1975) was the first who extended the economic order quantity model, incorporating inflation and time value of money. As other notable papers in this direction, mention should be made of the works by Beirman and Thomas (1977), Misra (1975), Misra (1979), Aggarwal (1981), Chandra and Bahner (1985), and Sana (2010c), among others. The current study is to model a combined project of fishery and poultry of birds. It is rational that the deteriorated (non-conforming quality) fish (mainly shrimp and other small fishes) are used as nutrient of poultry. Conversely, the excreta of birds, dead birds and poultry processing byproducts such as chicken bones, intestines, and whole carcasses are used as nutrient of fishery, after conversion of poultry litter. Consequently, the nutrients of fishery and poultry are interconnected. The growth rates of fish and birds in poultry are considered as

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S.S. Sana / Economic Modelling 29 (2012) 1024–1034

(γx, γy) food values per unit mass of nutrients of (θxX, θyY) respectively. (N01, N02) supplied nutrients, from outside, per unit biomass of fishes and birds of poultry respectively (αx, αy) conversion factors, both are positive, of nutrients form wastes of byproducts of fish processing and poultry wastes respectively (τx, τy) absorbed nutrients per unit biomass of (X, Y) respectively (Cx, Cy) cost of fermentation of poultry litter and fish-processing wastes (C1, C2) cost of unit mass of supplied nutrients (N01, N02) respectively (C3, C4) cost of medicine per unit mass of (X, Y) respectively (pxmax, pymax) maximum selling price per unit mass of fish and poultry birds (px(t), py(t)) selling price per unit mass of fishes and birds of poultry respectively δ = (r − i) r and i are rates of interest and inflation per unit currency

functions of available nutrients, volume of on-hand biomass and environmental carrying capacities simultaneously. The harvesting rates vary with selling prices and on-hand stock of the biomasses those adjust the demand of the customers in the market. To control the productivity of the system, baby species of breeding systems are introduced if needed. 2. Notation The following notations are considered to develop the model: Notation X(t) Y(t) X_ ðt Þ Y_ ðt Þ Nx(t) (κx, κy) Ny(t) (Lx, Ly) (rx, ry) θx

biomass of fish at time ‘t’. biomass of poultry birds (broiler/duck) at time ‘t’. denotes the derivative of X with respect to time ‘t’. denotes the derivative of Y with respect to time ‘t’. amount of nutrient at time ‘t’ for fish. positive constants. amount of nutrient at time ‘t’ for broiler. environmental carrying capacities of (X, Y) respectively biotic potential of (X, Y) respectively. deterioration rate which is a fraction of on-hand biomass of X(t) , and it is a function of Y. θy deterioration rate which is a fraction of on-hand biomass of Y(t) , and it is a function of X. (βx, βy) amount of waste per unit mass of (X, Y) respectively.

3. Formulation of the model The model considers a joint project of fishery and poultry. Quite often, the excreta of birds of poultry, dead birds, poultry processing by products and living birds are used as nutrient (food) of fish. By the by, the deteriorated fish ,shrimp and small fishes which have less market value are used as nutrient (food) of the broiler (see Fig. 1).

Harvesting

Supplied nutrients from outside

Supplied nutrients from outside Fermentation

Nutrients of poultry Nutrients of fishery

Fishery

Food processing

Fermentation

Wastes of poultry processing

Wastes of fish processing and less valued fishes

Food processing Market

Medicine

Harvesting

Poultry

Excreta of birds

Non-conforming quality birds

Fig. 1. Pictorial representation of the project.

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

The state of nutrients of (X, Y) at time ‘t’ are as follows: N x ðt Þ ¼ N01 X ðt Þ þ α y βy Y ðt Þ þ γ y θy Y ðt Þ−τ x X ðt Þ ¼ N1 X ðt Þ þ α y βy Y ðt Þ þ γ y θy Y ðt Þ; where N 1 ¼ N01 −τ x

1027


 sufficient amount, then growth rate tends to r x 1− LX . When X → Lx, x

ð1Þ

i.e., biomass of X reaches to environmental carrying capacity, then growth rate of X tends to zero which is rational in practice. Similarly,
 when Ny → ∞ , the growth rate of Y tends to r y 1− LY Y and the growth y

rate tends to zero when Y → Ly. The rates of consumers' demand (Dx, Dy) of the species (X, Y) are considered as follows:

and N y ðt Þ ¼ N02 Y ðt Þ þ α x βx X ðt Þ þ γ x θx X ðt Þ−τy Y ðt Þ ¼ N2 Y ðt Þ þ α x βx X ðt Þ þ γ x θx X ðt Þ; where N2 ¼ N 02 −τ y :

Dx ¼ ð2Þ

ð7Þ

! py 1− max Y py

ð8Þ

and



 ay x , θx ¼ 1− ay þY are deteriorated fractions of Here, θy ¼ 1− axaþX on-hand stock of X and Y species respectively. Also, ax and ay are positive constants. θy(0b θy b 1) is an increasing function of X because huge biomass of X may be used as direct nutrient of Y species. Similarly, θx(0b θx b 1) is an increasing function of Y species. N01 is supplied nutrient per unit mass of X from outside, αyβyY(t) is converted nutrient from poultry waste excluding dead birds, γyθyY (including dead and living birds) is direct nutrient of X and τx is the absorbed nutrient per unit mass of X. N02 is supplied nutrient per unit mass of poultry, (αxβx + γxθx)X is converted nutrient from fish processing by products; deteriorated and less valued fishes and τy is absorbed nutrient per unit mass of poultry birds. Substituting the values of θx and θy in Eqs. (1) and (2), we have   γ y ax Y ðt Þ; where δy ¼ α y βy þ γy Nx ðt Þ ¼ N1 X ðt Þ þ δy Y ðt Þ− ax þ X ðt Þ

  px 1− max X px

Dy ¼

where px and py, control variables, are the selling prices per unit of the two species(X, Y). We have 



X_ Y_

 ¼

G1 ðX; Y Þ G2 ðX; Y Þ

 G1 ðX; Y Þ ¼ r x 1−

½

ð3Þ

  X 1− X−Dx −θx X Lx κx



κ x þ N1 X þ δy Y−γ y

!   ay px − 1− − 1− max px ay þ Y

γ x ay X ðt Þ; where δx ¼ α x βx þ γ x : ð4Þ ay þ Y ðt Þ

G2 ðX; Y Þ ¼ r y

½

κy 1− κ y þ Ny 0

B B ¼ Y ry B B1− @



ð5Þ − 1−

and

ax ax þ X

1   C X  C A 1− L x Y



and

respectively. The governing dynamical system of the species (X, Y) is as follows:   Nx X 1− X−Dx −θx X Lx κ x þ Nx    κx X ¼ r x 1− 1− X−Dx −θx X Lx κ x þ Nx

0

κx κ x þ Nx

B ¼ X rx B @1−

!

X_ ðt Þ ¼ r x

ð9Þ

where

and

Ny ðt Þ ¼ N 2 Y ðt Þ þ δx X ðt Þ−



!

! Y 1− Y−Dy −θy Y Ly κy

κ y þ N2 Y þ δx X−γ x

ay ay þ Y

1 ! C C Y C ! C 1− Ly A X



!   py ax : − 1− pmax ax þ X y

For non-zero critical points ðx; y Þ, the following conditions are satisfied:

Y_ ðt Þ ¼ r y ¼ ry

! Y 1− Y−Dy −θy Y Ly ! ! κy Y 1− 1− Y−Dy −θy Y Ly κ y þ Ny Ny κ y þ Ny


where rx 1− κ

κx

x þN x

!




0 r x @1− ð6Þ

 1− LX X is the growth rate of X that depends upon x

existing nutrient (Nx) , environmental carrying capacity and on-hand

 κy 1− LY Y is the growth rate of Y that depends biomass of X, r y 1− κ þN y

y

κ x þ N 1 x þ δy y −γy

1   x  A 1− ax Lx  a þx y




ð10Þ

x

!   ay px ¼ 1− max þ 1− px ay þ y 0 r y @1−

y

upon existing nutrient (Ny), environmental carrying capacity and onhand biomass of Y. When Nx → ∞, i.e., available nutrient is more than

κx

¼

κy

1

y
 A 1− ay L y κ y þ N 2 y þ δx x−γx a þy x y !   py ax : 1− max þ 1− py ax þ x

! ð11Þ

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S.S. Sana / Economic Modelling 29 (2012) 1024–1034

Solving the Eqs. (10) and (11), we have the critical points of the dynamical system. Now, differentiating G1(X, Y) and G2(X, Y) partially with respect to ′ X′ and ′ Y′, we have 0

B ∂ ðG ðX; Y ÞÞ ¼ r x B @1− ∂X 1

κx κ x þ N 1 X þ δy Y−γy

ay − 1− ay þ Y

−

1 Lx

1−

!

½f

þ X rx



1     C X px  C A 1− L − 1− pmax x x Y

3.3. Local stability analysis

g

;

8 9 2 3 > > > > >  > < = 6 7 δ −γ a = ð a þ X Þ a ∂ X y x y x y 6 7 ðG ðX; Y ÞÞ ¼ X 6r x κ x 1−    2 −
2 7; > 4 5 Lx > ∂Y 1 ax > > a þ Y > > y Y ; : κ x þ N 1 X þ δy Y−γ y ax þ X 8 9 2 3 > > > >
 > > > !> 6 7 > > = 6 7 δx −γx ay = ay þ Y ∂ Y < ax 6 7 ðG2 ðX; Y ÞÞ ¼ Y 6r y κ y 1− ! !2 − 7; 2 > > ðax þ X Þ 7 6 Ly > ∂X > a > > y 4 5 > > þ N Y þ δ X−γ κ X > > 2 x x : y ; ay þ Y 0 1 ! ! B C B C κy py ∂ Y ! C ðG2 ðX; Y ÞÞ ¼ r y B 1− − 1− max 1− B C py Ly ∂Y ay @ A X κ y þ N 2 Y þ δx X−γx ay þ Y 0 1  γx ay X C
B κ y @N 2 þ
2 A 1−Y=Ly   ay þ Y ax þ Y ry − 1−

2 ax þ X κ y þ N2 Y þ δx X−γ x ay X= ay þ Y

½f

0 1 κy 1@ A
 − 1− Ly κ y þ N 2 Y þ δx X−γ x ay X= ay þ Y

∂G1 ∂X

 ðx ;y Þ

½f

¼ x r x





∂G1 ∂Y

∂G2 ∂X

∂G2 ∂Y

 ðx ;y Þ

 ðx ;y Þ

 ðx ;y Þ

1 Lx

½

½f

¼ y r y

0

∂X

ðG1 ðx; y ÞÞ þ

½f

¼ x r x

−

g

0 1 κy 1@
A − 1− Ly κ y þ N2 y þ δx x−γx ay x= ay þ y


2 κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ

1 Lx

1−

½f

0

1  γx ay x C
B κ y @N 2 þ
2 A 1−y =Ly ay þ y

2 κ y þ N2 y þ δx x−γ x ay x= ay þ y



g

:

3.1. Equilibria of the system The possible steady states of the dynamical system (9) are the solutions, ðx; y Þ, of Eqs. (10) and (11).

g

    ∂G2 ∂G1 ∂G2 − ∂Y ∂Y ∂X   γ y ax y κ x N1 þ ð1−x=Lx Þ 2 1 ðax þ xÞ ¼ x r x
2 − Lx κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ ! κx × 1− κ x þ N 1 x þ δy y −γy ax y =ðax þ xÞ

ψ2 ðx; y Þ ¼



g

! κx κ x þ N 1 x þ δy y −γy ax y =ðax þ xÞ

0 1 κy 1@
A 1− Ly κ y þ N 2 y þ δx x−γ x ay x= ay þ y

∂G1 ∂X



½f

y ry

g

0

1  γ x ay x C
B κ y @N 2 þ
2 A 1−y =Ly ay þ y 1

2 − Ly κ y þ N 2 y þ δx x−γ x ay x= ay þ y

0 ×@1−

y

1  γx ay x C
B κ y @N 2 þ
2 A 1−y =Ly ay þ y

2  κ y þ N 2 y þ δx x−γ x ay x= ay þ y

  γy ax y ð1−x=Lx Þ κ x N1 þ 2 ðax þ xÞ

þy r y

and

∂ ðG ðx; y ÞÞ ∂Y 2

½f

!

1−

ψ1 ðx ; y Þ ¼

:

  γ y ax y κ x N1 þ ð1−x=Lx Þ 2 ðax þ xÞ

2 κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ

0 where ∂  

g

κx ; κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ 8 9 > > > > >  > < =  δy −γ y ax =ðax þ x Þ ay x ¼ x r x κ x 1−    2 −
2 ; > Lx > a > > x a þ y > >    y x þ δ y −γ κ þ N y : x ; 1 y y  ax þ x 8 9 3 2 > > > >
 > > > 7 6 !> > > = 7 6 δx −γ x ay = ay þ y y < ax 7 6 ¼ y 6r y κ y 1− ! !2 − 7; 2 > > ðax þ xÞ 7 6 Ly > > a y > > 5 4 > > > : κ y þ N 2 y þ δx x−γx a þ y x > ; −






We shall now investigate the local behavior of critical points of the dynamical system in Eq. (9). The variational matrix of the system of Eq. (9) is 0 1 ∂ ∂     ð ð Þ Þ ð ð Þ Þ x ; y x ; y G G B C ∂X 1 ∂Y 1 C V ðx; y Þ ¼ B ð12Þ @ ∂ A ∂     ðG2 ðx ; y ÞÞ ðG2 ðx ; y ÞÞ ∂X ∂Y
 ðx ;y Þ ðx ;y Þ þ þ ∂G2∂Y The characteristic equation of V ðx; y Þ is λ2 −λ ∂G1∂X



 ∂G1 ðx ;y Þ ∂G2 ðx ;y Þ ∂G1 ðx ;y Þ ∂G2 ðx ;y Þ 2     , or, λ −ψ ð x ; y Þλ þ ψ ð x ; y Þ ¼ − ¼ 0 1 2 ∂X ∂Y ∂Y ∂X

−

At the non-zero critical point ðx; y Þ, the above partial derivatives are as follows: 

Lemma. All solutions of Eq. (9) which start in R2+ are uniformly bounded. Proof. See Appendix A.

ax ax þ X   γy ax Y κ x N1 þ ð1−X=Lx Þ 2 ðax þ X Þ
2 κ x þ N1 X þ δy Y−γy ax Y=ðax þ X Þ

! κx κ x þ N 1 X þ δy Y−γy ax Y=ðax þ X Þ

3.2. Boundedness of the system

κy

g

1


A κ y þ N 2 y þ δx x−γ x ay x= ay þ y 8 > > > <

9 > > >  = δy −γy ax =ðax þ xÞ ay x  −x rx κ x 1− 2    2 −
> Lx > a > > x ay þ y > y > : κ x þ N 1 x þ δy y −γ y ; ax þ x 8 9 3 2 > > > >
 > > > > 7 6 !> > = 7 6 δx −γ x ay = ay þ y ax y < 7 6 y 6ry κ y 1− ! !2 − 7 2 > 6 Þ 7 Ly > ð a þ x > > a x y > > 5 4 > >    κ y þ δ x −γ þ N x > > 2 x x : y ;  ay þ y

½



Now, ðx; y Þ to be a stable node if both the eigenvalues of the above is negative, i.e., ψ1 ðx; y Þb0 and ψ2 ðx; y Þ > 0

ð13Þ

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

are satisfied. In example 1, (x ¼ 607:11; y ¼ 467:58) is a stable node because the eigenvalues of the characteristic equation is negative (− 20.4142, − 0.34844) (see Fig. 2)

1029

800

3.4. Global stability analysis

600

Poultry

We shall study the global stability of the system of Eq. (9) by considering a suitable Lyapunov function F ðX; Y Þ ¼ ½ðX−xÞ−x ln ðX=xÞ þ h½ðY−y Þ−y lnðY=y Þ where h is a suitable constant. F ðx; y Þ is zero at the equilibrium point ðx; y Þ and is positive for all other values of (X, Y)  R2+. The time derivative of F along the trajectories of Eq. (9) is F_ ¼

200

    X−x _ Y−x _ X þh Y X Y 2 0

0

1 3 !     C 7 ay X p x C 1− 7   A − 1− max − 1− ax Lx px ay þ Y 5 Y κ x þ N 1 X þ δy Y−γ y ax þ X 1 2 0 3 ! !  C 6 B 7 C 6 B 7 κy p B C 1− Y − 1− y − 1− ax 7 ! C þhðY−y Þ6 6r y B1− Ly pmax ax þ X 7 ay y A 4 @ 5 X κ y þ N2 Y þ δx X−γ x ay þ Y 6 B B ¼ ðX−xÞ6 4r x @1−

400

κx

0

200 Fish

400

600

800

Stable Node

Fig. 2. Phase portrait of Example 1.

¼ ½X−x ; Y−y  P ½X−x; Y−y  T

3.5. Optimal harvesting policy where The net profit of the project, including inflation and time value of money, is



 a11 a12 ; P¼ a21 a22   γy ax y rx κ x r − x a11 ¼ N1 − AðX; Y ÞAðx; y Þ Lx C ðX ÞC ðxÞ  
 γ y ax κ x rx ððax þ xÞy þ X Þ ; κ x þ δy −γ y y þ þ L AðX; Y ÞAðx; y Þ C ðX ÞC ðxÞ "x   γy ax ðax þ xÞ 1 κ x rx a12 ¼ δy −γ y þ    C ðX ÞC ðx Þ 2 AðX; Y ÞAðx ; y Þ 
  γ y ax ðax þ 1Þx κ x rx þ − δy −γ y x− Lx AðX; Y ÞAðx; y Þ C ðX ÞC ðxÞ
1 0 γ x ay ay þ y ay κ y ry @δx −γ þ A − þh x   C ðY ÞC ðy Þ BðX; ðy Þ
C ðYÞC1 0 Y ÞBðx ; y Þ γ x ay ay þ 1 y κ y ry @−ðδx −γ x Þy − A þ   C ðY ÞC ðy Þ Ly BðX; Y ÞBðx ; y Þ −


 ∞ −δt J ¼ ∫0 π X; Y; px ; py e dt where

ax  ¼ a21 ; C ðX ÞC ðxÞ

  ry κ y γ x ay x ry N2 − − BðX; Y ÞBðx;y Þ Ly C ðY ÞC ðy Þ   κ y ry γ x ay

þ ; κ y þ ðδx −γ x Þx þ ay þ y x þ Y Ly BðX; Y ÞBðx; y Þ C ðY ÞC ðy Þ
 γ y ax Y AðX; Y Þ ¼ κ x þ N1 X þ δy −γ y Y þ ; ax þ X

½

a22 ¼ h

BðX; Y Þ ¼ κ y þ N2 Y þ ðδx −γ x ÞX þ

ð14Þ



γ x ay X ; ay þ Y

C ðX Þ ¼ ax þ X; C ðY Þ ¼ ay þ Y:

The eigenvalues of the characteristic equation of the above matrix are both negative if a11 + a22 b 0 and a11a22 − (a12) 2 > 0 are satisfied. Therefore, the interior equilibrium point ðx; y Þ is globally asymptotically stable if the above inequalities hold simultaneously. The critical point (607.11, 467.58) of example 1 is globally stable (see Fig. 3) as the above inequalities hold.


 π X; Y; px ; py ¼ Income from sales the biomasses −cost of nutrient from outside −medicinal cost applied just after harvesting −cost of fermentation of poultry litter and fish wastes "  !  py px X þ py 1− max Y−C 1 N 01 X−C 2 N 02 Y−C 3 ¼ px 1− max px py ! ! ay py px ax X−C 4 max −1 þ Y−C x × max −1 þ px ay þ Y py ax þ X !#   ay X a Y × β y Y þ Y− x −C y β x X þ X− ay þ Y ax þ X    ay X px −C y ðβ x þ 1Þ−C 1 N 01 X−C 3 ¼ ðpx þ C 3 Þ 1− max ay þ Y px ( ! )

 py a Y þ py þ C 4 1− max −C x βy þ 1 −C 2 N 02 Y−C 4 x py ax þ X

Now our objective is to maximize J subject to the system of Eq. (9), using Pontryagin's maximum principle. The control variables px(t) and py(t) are subject to the constraints 0 ≤ px(t) ≤ pxmax and 0 ≤ py(t) ≤ pymax, pxmax and pymax are feasible upper limits for the selling prices. The Hamiltonian of the problem is "  
 ay X px H X; Y; px ; py ¼ ðpx þ C 3 Þ 1− max −C y ðβx þ 1Þ−C 1 N 01 X−C 3 px ay þ Y # ( ! )

 py aY −δt e þ py þ C 4 1− max −C x β y þ 1 −C 2 N 02 Y−C 4 x ax þ X py þλ1 ðt ÞG1 þ λ2 ðt ÞG2

ð15Þ

1030

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

510 611

Biomass of Fishery

500

Poultry

y

490 480 470

Fish

610

609

1

2

3

4

5

460 607

450 590

600

610

620

630

640

Time

650

x

Fig. 4. Biomass of fishery (X) versus time.

Fish and

Fig. 3. Global attractor of the Example 1.

where λ1(t) and λ2(t) are adjoint variables. The optimal control variables px(t) and py(t) which maximize H, must satisfy the following conditions:

Q 2 ¼ δ−x r x
2 κ x þ N 1 x þ δy y −γy ax y =ðax þ xÞ

∂H ¼0 ∂px ∂H ¼0 ∂py dλ ∂H − 1¼ dt ∂X dλ ∂H − 2¼ : dt ∂Y

−




2 max  Q 1 ¼ δ −ψ1 δ þ ψ2 2px þ C 3 −px ;

ð16Þ



 2 max Q 2 ¼ δ −ψ1 δ þ ψ2 2py þ C 4 −py

ð17Þ

0

1  γ x ay x C
B κ y @N 2 þ
2 A 1−y =Ly ay þ y

2 κ y þ N 2 y þ δx x−γ x ay x= ay þ y

þy r y κ y 1−

½





1−

½

!
 py ay x ax þ C3
−C x βy þ 1 −C 2 N 02 −C 4 2 pmax ax þ x y ay þ y

½







Example 1. We consider the values of the parameters in appropriate units as follows:κx = 4, κy = 4, N01 = 1.5, N02 = 1.5, rx = 10.0, ry = 12.0, Cx = 0.8, Cy = 0.6, Lx = 650, Ly = 500, ax = 2000, ay = 2500, αx = 0.2, αy = 0.5, βx = 0.3, βy = 0.6, γx = 11, γy = 10, τx = 3.5, τy = 3.8,

g

335

  ay px ax y þ C4  ðpx þ C 3 Þ 1− max −C y ðβx þ 1Þ−C 1 N01 −C 3 px ay þ y ðax þ xÞ2

½

py þ C 4

g

!

  ay px ax y þ C4  ðpx þ C 3 Þ 1− max : −C y ðβx þ 1Þ−C 1 N 01 −C 3 px ay þ y ðax þ xÞ2

0 1 κy 1@
A 1− Ly κ y þ N 2 y þ δx x−γ x ay x= ay þ y

½

κx κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ

8 9 > > > > >  > < =  δy −γy ax =ðax þ x Þ ay x þx r x κ x 1− 2    2 −
> Lx > a > > x  a > >    y þy y ; : κ x þ N 1 x þ δy y −γ y ax þ x

8 > > > !> > y <

9 > > > > > = ax ! !2 − > ð a þ xÞ2 > ay x > x > > ; ay þ y


 δx −γ x ay = ay þ y

Ly > > > > > : κ y þ N2 y þ δx x−γx



Growth Rate of Fishery

−




1−

Now, solving Eqs.
(10), (11), (16) and (17), we have the optimal equilibrium solution x; y ; p x ; p y . The following numerical example is used to justify the proposed model.

where

½ ½f

1 Lx

½



Solving the above equations (see Appendix B), we have

Q 1 ¼ δ−y r y

½ ½f

  γ y ax y κ x N1 þ ð1−x=Lx Þ ðax þ xÞ2



!

 py ay x ax þ C3
py þ C 4 1− max −C x βy þ 1 −C 2 N 02 −C 4 2 py ax þ x ay þ y



330

325

1

2

3

Time Fig. 5. Growth rate of fish (X) versus time.

4

5

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

1031

221.5

221.0

1

2

3

4

5

Time

Growth Rate of Poultry Birds

Harvested Rate of Fish

280

270

260

250

1

Fig. 6. Harvested rate of fish (X) versus time.

2

3

4

5

Time pymax = $55,

C1 = $3.0, C2 = $2.0, C3 = $2.0, C4 = $3.0, r = 16 %, i = 11 %, δ = 0.05. Then the optimal solution ðx ¼ 607:11; y ¼ 467:58; px ¼ 31:87; py ¼ 34:95Þ is locally as well as globally stable node (see Figs. 2 and 3) because the eigenvalues (−20.4142, − 0.34844) are negative. The on-hand stocks, growth rates, harvesting rates, available nutrients of the fishery and poultry at time ‘t’ are shown in the following figures (see Figs. 4 to 11) 4. Conclusion Poultry litter is an excellent substrate for the growth of microorganisms for it allows the development of a dense population of protozoa. Also, it has a high content of nitrogen and phosphorus and biochemical oxygen demand (BOD). The organisms (phytoplankton, detritus, microorganisms such as bacteria and protozoa) can consume poultry litter directly and they can grow on the natural feed that contributes to the development of the nutrient of fish. The timely availability of replacement stock, veterinary support, and market demand may be critical to maintaining both poultry, and their waste, production. Higher loadings of waste necessitate water exchange or mechanical aeration to maintain dissolved oxygen. Overloading of poultry waste can also be avoided by housing poultry over concrete or earthen floors rather than directly over ponds and regular manual or mechanical collection and addition. This option may reduce construction costs considerably and also enables farmers to sell manure surplus to their requirements. On the other hand, fish protein is used extensively in poultry and swine production as a source of high quality protein. Waste products from fish-processing operations and the non-conforming quality species of fish are used as fish protein for animal feed. The present model is an integrated project of fishery and poultry. The motivation behind the concept is the use of deteriorated fish (mainly shrimp, small fishes, low valued fishes, etc.) as the nutrient for poultry and after conversion of poultry litter, the excreta of birds, dead birds and byproducts of poultry processing are used as nutrient of fishery. Moreover, the selling prices of both the species (poultry and fishery) are

Fig. 8. Growth rate of poultry birds (Y) versus time.

control variables which are controlled by maximizing the joint profits of the project. The demand of the species in the market is controlling the sales prices and stocks of the species by the farm manager. From the analysis of the model, the following factors can be summarized as follows 1. The growth rates of fish and birds in poultry are considered as functions of available nutrients, environmental carrying capacity and volume of on-hand biomass simultaneously. Both of the species (fishery and poultry) are governed by the logistic law of growth which is new in this literature. 2. The demands of the species in the market are functions of sales price and on-hand stock of the project. 3. The existence of local stability and the global stability validate the model and the phase portrait also shows that it is a node. It confirms the fact that, the nutrients of fishery and poultry are interconnected. 173.0

Harvested Rate of Poultry Birds

pxmax = $50,

172.5

172.0

171.5

171.0

1

2

3

4

5

Time Fig. 9. Harvested rate of poultry birds (Y) versus time.

474

Biomass of Poultry

472 471 470 469

1

2

3

4

5

Available Nutrients of Fishery

30 473

25

20

15

Time Fig. 7. Biomass of poultry (Y) versus time.

1

2

3

4

Time Fig. 10. Available nutrient of fishery (x) versus time.

5

1032

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

Available Nutrients of Poultry Birds

For each s > 0, we obtain   1 1 U_ þ sU ¼ X_ þ Y_ þ s X þ Y ℓ ℓ

18

½

17

¼ X r x 1−

16

!  κx X 1− κ x þ N1 X þ δy Y−γy ax Y=ðax þ X Þ Lx



!   ay px − 1− max − 1− px ay þ Y

15

½

1 κy 1
A þ Y r y @1− ℓ κ y þ N2 Y þ δx X−γx ay X= ay þ Y

14

1

2

3

4

0



! !     py Y ax 1 − 1− max − 1− þs Xþ Y Ly ℓ py ax þ X !     py px 1 1 b X r x − 1− max þ ay þ Y r y − 1− max þ ax þ s X þ Y ℓ ℓ px py

5

Time

½

Fig. 11. Available nutrient of poultry birds (Y) versus time.

 1−

½ ½

 ½  ½  ½



4. The optimal harvest policy is analyzed invoking Pontryagin's maximal principle, subject to the state equations and the control constraints. The optimal equilibrium solution is obtained by a suitable and realistic numerical example. It is found that the shadow price remains constant over time in optimal equilibrium when they satisfy the transversality condition.

!    py px Y
¼ X ðr x þ sÞ− 1− max þ ay þ r y þ s − 1− max þ ax ℓ px py

The present model is unique in many ways. The unique features of the model are outlined below

b K; since 0 ≤ X ≤Lx and 0 ≤ Y ≤ Ly

(i) The relationship between the fishery and poultry through nutrients. (ii) The demand rates of both the species are new in the recent research works in this field. (iii) Concept of inflation and time value of money of cost and profit parameters are considered. The present article may be extended further for stochastic differential equations in which the key parameters are stochastic in character. The age structures of the fishes and poultry birds with and without delays of harvesting, in which the stability analysis is more complicated, are left for future extension. Appendix A

 

!    Ly
py px þ ay þ b Lx ðr x þ sÞ− 1− max ry þ s − 1− max þ ax px ℓ py

where "  ! ##    "
 Ly py px K=2 ¼ Max Lx ðr x þ sÞ− 1− max þ ay ; r y þ s − 1− max þ ax : px ℓ py

Applying the theory of differential inequality (Birkhoff and Rota, 1982), we have
 −st −st þ U ðX ð0Þ; Y ð0ÞÞe : 0bU ðX; Y ÞbðK=sÞ 1−e

ð19Þ

When t → ∞, the above yields 0 b U b K/s. Therefore, all the solution of Eq. (9) that starts in R2+ confined to the region R where n þ R ¼ ðX; Y ÞR2 : U ¼ ðK=sÞ þ ; forany> 0g:

Lemma. All solutions of Eq. (9) which start in R2+ are uniformly bounded.

Appendix B

Proof. Let the function 1 Y: ℓ The time derivative of Eq. (12) is

1 U_ ¼ X_ þ Y_ ℓ

!  κx X ¼ X r x 1− 1− Lx κ x þ N1 X þ δy Y−γy ax Y=ðax þ X Þ !   ay px − 1− − 1− max px ay þ Y

½

0


 Now, ∂ H/∂ px = 0 and ∂ H/∂ py = 0 at x; y ; p x ; p y gives us

ð18Þ

U ðX; Y Þ ¼ X þ

½

Hence the proof. □



1 κy 1
A þ Y r y @1− ℓ κ y þ N2 Y þ δx X−γx ay X= ay þ Y ! !   py Y ax  1− : − 1− max − 1− Ly py ax þ X



max  −δt λ1 ðt Þ ¼ 2px þ C 3 −px e ;

ð20Þ


 max −δt e : λ2 ðt Þ ¼ 2py þ C 4 −py

ð21Þ


 1 ¼ ∂H at x; y ; p x ; p y gives us Now, − dλ dt ∂X

−

      dλ1 ∂π ∂G1 ∂G2 −δt ¼ þ λ1 e þ λ2 : dt ∂X ðx;y ;p x ;p y Þ ∂X ðx;y ;p x ;p y Þ ∂X ðx;y ;p x ;p y Þ ð22Þ Similarly,

−

      dλ2 ∂π ∂G1 ∂G2 −δt ¼ þ λ1 e þ λ2 :ð23Þ dt ∂Y ðx;y ;p x ;p y Þ ∂Y ðx;y ;p x ;p y Þ ∂Y ðx;y ;p x ;p y Þ

S.S. Sana / Economic Modelling 29 (2012) 1024–1034

where

From Eq. (22), we have       dλ1 ∂π −δt ∂G ∂G2 þ þ λ1 1 = : e λ2 ¼ − dt ∂X ∂X ∂X

ð24Þ

½

½ ½f

          ∂G1 ∂G2 ∂G1 ∂G2 ∂G1 ∂G2 þ þ λ1 − ∂X ∂Y ∂X ∂Y ∂Y ∂X          ∂π ∂G2 ∂π ∂G2 ∂π −δt ¼ δ þ − e ∂X ∂Y ∂X ∂Y ∂X −δt

¼ Q 1e 2

ð25Þ 

where

½



∂π ∂X

Q1 ¼ δ



½ ½f

¼ δ−y ry



ðx;y ;p x ;p y Þ 1
  γ x ay x C B κ y @N 2 þ
2 A 1−y =Ly ay þ y

2 κ y þ N 2 y þ δx x−γ x ay x= ay þ y

−

  ay px ax y þ C4  ðpx þ C 3 Þ 1− max −C y ðβx þ 1Þ−C 1 N 01 −C 3 px ay þ y ðax þ xÞ2

½

þy r y κ y 1−



½

8 > > > !> > y <

9 > > > > > = ax ! !2 − > ð a þ xÞ2 > ay x > x > > ; ay þ y


 δx −γ x ay = ay þ y

Ly > > > > > : κ y þ N 2 y þ δx x−γx





!

 py ay x ax þ C3
py þ C 4 1− max −C x βy þ 1 −C 2 N 02 −C 4 2 py ax þ x ay þ y

2



ð26Þ

The roots (μ1, μ2) of Eq. (26) are positive by virtue of Eq. (13). Therefore, the solution of λ1(t) is μ1t

λ1 ðt Þ ¼ Ae

þ Be

μ2t

þ

Q1 −δt e : δ2 −ψ1 δ þ ψ2

The shadow price λ1(t)e δt remains bounded as t → ∞ if and only if A = 0 = B and then λ1 ðt Þ ¼

Q1 −δt e : δ2 −ψ1 δ þ ψ2

ð27Þ

Similarly, we have λ2 ðt Þ ¼

Q2 −δt e : δ2 −ψ1 δ þ ψ2

! κx κ x þ N 1 x þ δy y −γ y ax y =ðax þ xÞ

g

!

 py ay x ax þ C3
py þ C 4 1− max −C x β y þ 1 −C 2 N 02 −C 4 2 py ax þ x ay þ y

½










2 max  Q 1 ¼ δ −ψ1 δ þ ψ2 2px þ C 3 −px ;

ð29Þ



 2 max Q 2 ¼ δ −ψ1 δ þ ψ2 2py þ C 4 −py

ð30Þ

References

and (ψ1, ψ2) are as before. The auxiliary equation of Eq. (25) is μ þ ψ1 μ þ ψ2 ¼ 0:

1−

Substituting λ1(t) and λ2(t) from Eqs. (27) and (28) in Eqs. (20) and (21), we have

1

κy 1@
A 1− Ly κ y þ N 2 y þ δx x−γx ay x= ay þ y

½

½

½

g

0

1 Lx

9 8 > > > > >  > = δy −γ y ax =ðax þ xÞ ay x <  þx r x κ x 1− 2    2 −
> Lx > a > > x ay þ y > y > ; : κ x þ N 1 x þ δy y −γy ax þ x   ay px ax y −C y ðβ x þ 1Þ−C 1 N 01 −C 3  ðpx þ C 3 Þ 1− max þ C4 px ay þ y ðax þ xÞ2

      ∂G2 ∂π ∂G2 ∂π þ − ∂X ∂Y ∂Y ∂X 0

ðx;y ;p x ;p y Þ

  γ y ax y κ x N1 þ ð1−x=Lx Þ 2 ðax þ xÞ

¼ δ−x r x
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−

d λ dλ −δt i:e:; 21 þ ψ1 1 þ ψ2 λ1 ¼ Q 1 e dt dt



         ∂π ∂G1 ∂π ∂G1 ∂π δ þ − ∂Y ∂X ∂Y ∂Y ∂X

Q2 ¼

Substituting the above in Eq. (23), we have d2 λ1 dλ1 þ dt dt 2

1033

ð28Þ

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