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Research article

Fuzzy optimal harvesting of a prey-predator model in the presence of toxicity with prey refuge under imprecise parameters


  • The objective of this paper is to investigate the dynamic behaviors of a prey-predator model incorporating the effect of toxic substances with prey refuge under imprecise parameters. We handle these biological parameters in model by using interval numbers. The existence together with stability of biological equilibria are obtained. We also analyze the existence conditions of the bionomic equilibria. The optimal harvesting strategy is explored by taking into account instantaneous annual discount rate under fuzzy conditions. Three numeric examples are performed to illustrate our analytical findings.

    Citation: Shuqi Zhai, Qinglong Wang, Ting Yu. Fuzzy optimal harvesting of a prey-predator model in the presence of toxicity with prey refuge under imprecise parameters[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 11983-12012. doi: 10.3934/mbe.2022558

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  • The objective of this paper is to investigate the dynamic behaviors of a prey-predator model incorporating the effect of toxic substances with prey refuge under imprecise parameters. We handle these biological parameters in model by using interval numbers. The existence together with stability of biological equilibria are obtained. We also analyze the existence conditions of the bionomic equilibria. The optimal harvesting strategy is explored by taking into account instantaneous annual discount rate under fuzzy conditions. Three numeric examples are performed to illustrate our analytical findings.



    Let A be an abelian category and X a contravariantly finite subcateogry of A. This means that for each object MA there exists a (not necessarily exact) complex X1X0M0, where XiX for every i0, which is exact after applying the functor HomA(X,) for each XX. Such a complex is called an X-resolution of M and denoted by XM, where X is the corresponding deleted complex. Since X is unique up to homotopy, so we can compute right derived functors of Hom, denoted by ExtnX. In many cases there is "balance'' in the computation of such functors, meaning that there exists a convariantly finite subcateogry Y of A such that ExtnX(M,N) can be also obtained from the right derived functors ExtnY computed from a Y-coresolution of N, 0NY0Y1, where YiY for every i0. This phenomenon can be summarized by saying that the pair (X,Y) is a balanced pair (in the sense of Chen [3]) or equivalently that the functor Hom is right balanced by X×Y (see Enochs and Jenda [5,Section 8.2]). In this case, we donote by Exti(M,N) the abelian groups ExtiX(M,N)ExtiY(M,N) (see [5]). Let P and I be classes of projective and injective left R-modules, respectively. Then the pair (P,I) is a classical example of balanced pairs. Balanced pairs have gained attention in the last years and it is very useful in relative homological algebra because balanced pairs are very closely related to resolutions, triangle-equivalences, cotorsion pairs and recollements (see for instance [3,5,6,12,18]).

    Let X be a subcategory of A. Sather-Wagstaff, Sharif and White introduced in [13] the Gorenstein subcategory G(X), which unifies the following notions: modules of G-dimension zero, Gorenstein projective modules, Gorenstein injective modules and so on. The Gorenstein subcategory G(X) of A is defined as G(X)={MAthereexistsanexactsequenceX=X1X0X1inX,whichisbothHomA(X,)exactandHomA(,X)exact,suchthatMker(X0X1)}. Such a complex X is called a complete X-resolution of M. Recently, Gorenstein subcategories have been extensively studied by many authors, see [2,11,13,16] for instance.

    Let (X,Y) be an admissible balanced pair of abelian category A. Then we have two Gorenstein subcategories: G(X) and G(Y). We can establish Gorenstein homological dimensions in terms of these categories: the G(X) dimension of an object AA, denoted by G(X)dimA, is the minimal integer n0 such that there is an X-resolution 0GnG0A0 with each GiG(X). If there is no such an integer, set G(X)dimA=. Dually, we can define the notion of G(Y) codimension of an object AA, denoted by G(Y)codimA. If A is the category of left R-modules and (X,Y)=(P,I), then G(X) dimension and G(Y) codimension are exactly Gorenstein projective dimension and Gorenstein injective dimension defined by Holm in [10], respectively. Denote by silp R (respectively, spliR) the supremum of injective (respectively, projective) dimensions of the projective (respectively, injective) modules (see [7]). Let ModR the category of left R-modules. The following is well known.

    Theorem 1.1. (see [4,Theorem 4.1]) Let R be a ring. Then the following are equivalent for any nonnegative integer n:

    (1) For any R-module M, G(P)dimMn;

    (2) For any R-module N, G(I)codimNn;

    (3) silp R=spli Rn.

    Moreover,

    sup{G(P)dimMMModR}=sup{G(I)codimMMModR}.

    In this case, we say that R has finite left Gorenstein global dimension and define the common value of these two numbers to be its left Gorenstein global dimension.

    The main goal of this paper is to generalize Theorem 1.1 to Gorenstein subcategories G(X) and G(Y) induced by any admissible balanced pair (X,Y) over an abelian category A, without assuming that A has projective or injective objects. The following is the main result of this paper:

    Theorem 1.2. Let (X,Y) be an admissible balanced pair over an abelian category A. Then the following are equivalent for any nonnegative integer n:

    (1) For any object A in A, G(X)dimAn.

    (2) For any object A in A, G(Y)codimAn.

    (3) sup{Y-cores.dimPPX}=sup{X-res.dimIIY}n.

    Moreover,

    sup{G(X)dimMMA}=sup{G(Y)codimMMA}.

    The common value of the last equality is called the Gorenstein global dimension of the abelian category A relative to (X,Y). What interests us is that if we consider the balanced pair induced by pure projective and pure injective modules over a ring R, then the induced Gorenstein global dimension is exactly the pure global dimension (see Corollary 2). If we consider the balanced pair induced by Gorenstein projective and Gorenstein injective modules over a Ding-Chen ring, then the induced Gorenstein global dimension is exactly the Gorenstein global dimension (see Corollary 2.9).

    The proof of the above results will be carried out in the next section.

    Throughout this section, we always assume that A is an abelian category (not necessarily with projective objects or injective objects). Let X be a full subcategory of A which is closed under taking direct summands. Let MA. A morphism f:XM is called a right X-approximation of M, if XX and any morphism g:XM with XX factors through f. The subcategory X is called contravariantly finite if each object in A has a right X-approximation. Dually one has the notion of left Y-approximation and then the notion of covariantly finite subcategories. We start by recalling the definition of balanced pairs.

    Definition 2.1. (see [3,Definition 1.1]) A pair (X,Y) of additive subcategories in A is called a balanced pair if the following conditions are satisfied:

    (BP0) the subcategory X is contravariantly finite and Y is covariantly finite;

    (BP1) for each object M, there is an X-resolution XM such that it is acyclic by applying the functors HomA(,Y) for all YY;

    (BP2) for each object N, there is a Y-coresolution NY such that it is acyclic by applying the functors HomA(X,) for all XX.

    We say that a contravariantly finite subcategory XA is admissible if each right X-appoximation is epic. Dually one has the notion of coadmissible covariantly finite subcategory. It turns out that X is admissible if and only if Y is coadmissible for a balanced pair (X,Y) (see [3,Corollary 2.3]). In this case, we say the balanced pair is admissible. In what follows, we always assume that (X,Y) is an admissible balanced pair in A.

    Lemma 2.2. If the short exact sequence 0XYZ0 is HomA(X,)-exact, then the following hold:

    (1) If ZG(X), then XG(X) if and only if YG(X). Moreover, G(X) is closed under direct summands.

    (2) If X,YG(X), then ZG(X) if and only if Ext1(Z,Q)=0 for any QX.

    (3) If XG(Y), then YG(Y) if and only if ZG(Y). Moreover, G(Y) is closed under direct summands.

    (4) If Y,ZG(Y), then XG(Y) if and only if Ext1(I,Z)=0 for any IY.

    Proof. We just prove (1) and (2) since (3) and (4) follow by duality.

    (1) The first statement follows from [1,Proposition 2.13(1)]. One can prove that G(X) is closed under direct summands by the proof similar to that of [17,Theorem 2.5].

    (2) The "only if" part is clear. For the "if" part, since XG(X), there exists a HomA(X,)-exact short exact sequence 0XPK0 with PX and KG(X). Then we have the following commutative diagram:

    sup{G(X)dimMMA}=sup{G(Y)codimMMA}.

    By [3,Propsotion 2.2], all rows and columns are HomA(X,)-exact. It follows from (1) that GX since Y,KX. By assumption Ext1(Z,P)=0, we know that the middle row in the above diagram splits. So ZG(X) by (1).

    Recall that the X-resolution dimension X-res.dimA of an object A is defined to be the minimal integer n0 such that there is an X-resolution

    sup{G(X)dimMMA}=sup{G(Y)codimMMA}.

    If there is no such an integer, set X-res.dimA=. Let YA be another full subcategory which is closed under taking direct summands. Dually one has the notion of Y-coresolution dimension Y-cores.dimB of an object B. For details, see [5,8.4]. We let ˜X (respectively, ˜Y) to denote the full subcategory of A whose objects are of finite X-resolution (respectively, Y-coresolution) dimension.

    Lemma 2.3. The following are true for any object AA:

    (1) If A˜X or A˜Y, then Exti(G,A)=0 for any i1 and GG(X).

    (2) If A˜X or A˜Y, then Exti(A,H)=0 for any i1 and HG(Y).

    Proof. We just prove (1) since (2) follows by duality. If GG(X), then it is easy to see that Exti(G,P)=0 for any PX and all i1. If A˜X, then there exists an X-resolution

    0PnPn1P1P0A0

    with all PiX. Hence Exti(G,A)Exti+n(G,Pn)=0 for any integer i1. Since GG(X), there exist a HomA(X,)-exact sequence

    0KjQj1Kj10

    where Qj1X and KjG(X) for any integer j0, here K0=G. Therefore Exti(G,A)Exti+1(K1,A)Exti+n(Kn,A). If A˜Y, it is easy to see that Exti(G,A)=0 for any integer i1, as desired.

    We let ˜G(X) (respectively, ˜G(Y)) to denote the full subcategory of A whose objects are of finite G(X) (respectively, G(Y)) (co)dimension.

    Proposition 2.4. The following are true for any 0AA:

    (1) If A˜G(X), then the following are equivalent:

    (i) G(X)dimAn;

    (ii)For any HomA(X,)-exact complex 0KnGn1G0A0 with all GiG(X), then so is Kn;

    (iii)Exti(A,Q)=0 for all i>n and all Q˜X;

    (iv)Exti(A,Q)=0 for all i>n and all QX.

    In this case, G(X)dimA=sup{iN0:Exti(A,Q)0 for some Q˜X}.

    (2) If A˜G(Y), then the following are equivalent:

    (i) G(Y)codimAn;

    (ii)For any HomA(X,)-exact complex 0AG0Gn1Kn0 with all GiG(Y), then so is Kn;

    (iii)Exti(I,A)=0 for all i>n and all I˜Y;

    (iv)Exti(I,A)=0 for all i>n and all IY;

    In this case, G(Y)codimA=sup{iN0:Exti(I,A)0 for some I˜Y}.

    Proof. We just prove (1) since (2) follows by duality.

    (i)(ii) It is obvious by a argument similar to that of [10,Proposition 2.7].

    (ii)(iii) Note that (X,Y) is a balanced pair. Then there exists a HomA(X,)-exact complex 0KnPn1P0A0 with all PiX. Hence KnG(X) by (ii) and Exti(A,Q)Extin(Kn,Q)=0 for all i>n and all Q˜X by Lemma 2.3(1).

    (iii)(iv) This is obvious.

    (iv)(i) Assume that G(X)dimA=m>n. Since (X,Y) is a balanced pair, there exists a HomA(X,)-exact sequence 0KmPm1P0A0 with all PiX and KmG(X) by hypothesis. Then there is a short exact sequence 0KjPj1Kj10 which is HomA(X,)-exact for each 1jm with K0=A. Hence Ext1(Km1,Q)Extm(A,Q)=0 for any QX by (iv). Consider the HomA(X,)-exact sequence 0KmPm1Km10 where Km,Pm1G(X), it follows from Lemma 2.2(2) that Km1G(X). Therefore G(X)dimAm1, a contradiction. So G(X)dimAn.

    Proposition 2.5. The following are true for any 0AA:

    (1) If G(X)dimA=n<, then there exist HomA(X,)-exact sequences

    0KGA0 and 0ALG0

    such that G,GG(X) and X-res.dimKn1 (if n=0, this should be interpreted as K=0) and X-res.dimLn.

    (2) If G(Y)codimA=n<, then there exist HomA(X,)-exact sequences

    0AGK0 and 0GLA0

    such that G,GG(Y) and Y-cores.dimKn1 (if n=0, this should be interpreted as K=0) and Y-cores.dimLn.

    Proof. According to Lemma 2.2(1), the results follows by an argument similar to that of Proposition 3.3 in [17].

    Corollary 2.6. The following are true for any object AA:

    (1) If A˜Y, then X-res.dimA=G(X)dimA.

    (2) If A˜X, then Y-cores.dimA=G(Y)codimA.

    Proof. We just prove (1) since (2) follows by duality. It is clear G(X)dimAX-res.dimA, so it is enough to show that X-res.dimAG(X)dimA, which is trivial when G(X)dimA=. Now assume G(X)dimA=n<, then there exists a HomA(X,)-exact sequence 0KGA0 such that GG(X) and X-res.dimK=n1 by Proposition 2.5(1). Since GG(X), there exists a HomA(X,)-exact sequence 0GPL0 such that PX and LG(X). Hence there exists the following commutative diagram:

    0AGK0 and 0GLA0

    where all rows and columns are HomA(X,)-exact. Hence X-res.dimMX-res.dimK+1=n by the middle row in above diagram. Note that A˜Y by hypothesis, Ext1(L,A)=0 by Lemma 2.3(1). So the third column in this diagram splits, and MLA. This implies that X-res.dimAX-res.dimMn=G(X)dimA, as desired.

    Proposition 2.7. If G(X)dimAn or G(Y)codimAn for any object AA and any positive integer n, then

    sup{Ycores.dimPPX}=sup{Xres.dimIIY}n.

    Proof. Suppose G(X)dimAn for any object AA. It is easy to see that sup{X-res.dimIIY}< by Corollary 2.6. Now assume that PX. For any object A, we have G(X)dimAn by hypothesis. It follows from Proposition 2.4(1) that Exti(A,P)=0 for all i>n. Hence Y-cores.dimPn, so sup{Ycores.dimPPX}n. In the following, we claim that if both sup{Ycores.dimPPX} and sup{Xres.dimIIY} are finite, then they are equal. Indeed, assume that sup{Ycores.dimPPX}=t and sup{Xres.dimIIY}=s. So there exists IY such that X-res.dimI=s. It is easy to check that there exists PX such that Exts(I,P)0, this implies that Y-cores.dimPs. Hence sup{Ycores.dimPPX}=ts. Similarly, one can get st. Then the result follows.

    Let 0KPA0 be a HomA(X,)-exact sequence with PX. Then object K is called a first syzygy of A, denoted by Ω1(A). An n-th syzygy of A, denoted by Ωn(A), is defined as usual by induction. Dually, one can define n-th cosyzygy of A, denoted by Ωn(A). We are now in a position to prove Theorem 1.2.

    Proof of Theorem 1.2. (1)(3) and (2)(3) follow from Proposition 2.7.

    (3)(1) First, we claim that any HomA(X,)-exact complex of objects in X is a complete X-resolution. Indeed, let T be a HomA(X,)-exact complex with each term in X. We show that for each integer n, the relevant exact sequence 0Kd+1TdKd0 is HomA(,X)-exact. If PX, then Y-cores.dimP=tn by hypothesis. Hence Ext1(Kd,P)Extt+1(Kdt,P)=0 because each TiX for any integer i, which implies that these HomA(X,)-exact sequences are HomA(,X)-exact. So T is a complete X resolution.

    Let A be an object in A. Assume that

    0AI0I1

    is a Y-coresolution of A and consider the HomA(X,)-exact sequences

    0KiIiKi+10,i0.

    Here K0=A. The following commutative diagram follows from the Horseshoe Lemma

    0KiIiKi+10,i0.

    with Ωn(Ki)Ωn(Ii)Ωn(Ki+1). Since sup{Xres.dimIIY}n, we have Ωn(Ii)X for any i0. Paste these HomA(X,)-exact sequences

    Ωn(Ki)Ωn(Ii)Ωn(Ki+1),i0,

    together, there is a HomA(X,)-exact complex

    0Ωn(A)Ωn(I0)Ωn(I1)

    with Ωn(Ii)X for any i0. Let P1P0A0 be an X-resolution of A. Similarly, one can get a HomA(X,)-exact complex

    Ωn(P1)Ωn(P0)Ωn(A)0

    with Ωn(Pi)X for any i0. Hence there is a HomA(X,)-exact complex

    Ωn(P1)Ωn(P0)Ωn(I0)Ωn(I1)

    with each term in X. By the claim before, we obtain that the complex above is a complete X-resolution of Ωn(A). So Ωn(A)G(X), and hence G(X)dimAn.

    Dually, one can prove (3)(2).

    The last equality is immediate from above equivalences and Proposition 2.7.

    Let R be a ring and ModR the category of left R-modules, and let PP(R) and PI(R) be the subcategories of Mod R consisting of pure projective modules and pure injective modules, respectively. Then (PP(R),PI(R)) is an admissible balanced pair in Mod R (see [5,Example 8.3.2]). In this case, G(PP(R)) coincides with pure projective modules. Indeed, any pure acyclic complex of pure projective modules is necessarily contractible. Dually, G(PI(R)) coincides with the pure injective modules. Denote by ppdRM (respectively, pidRM) the pure projective (respectively, pure injective) dimension of M. Recall that the pure global dimension of R is the supremum of pure projective dimensions of left R-modules, which is also equal to the supremum of pure injective dimensions of left R-modules (see [14,Section 2]). As a consequence of Theorem 1.2, we have the following result.

    Corollary 2.8. Let R be a ring. Then R has finite pure global dimension if and only if there exists a nonnegative integer n, such that

    sup{pidRPPPP(R)}=sup{ppdRIIPI(R)}n.

    Let R be a Ding-Chen ring (that is, a left and right coherent ring R such that both RR and RR have finite absolutely pure dimension). Then the pair (G(P),G(I)) is an admissible balanced pair in ModR (see [9,Theorem 1.1] and [8,Theorem 4.7]). In this case, by the stability of Gorenstein projective modules (see [13,Theorem A]), we see that G(G(P)) coincides with Gorenstein projective modules. Dually, G(G(I)) coincides with Gorenstein injective modules. Note that R has finite left Gorenstein global dimension if and only if sup{G(I)codimPPG(P)}=sup{G(P)dimIIG(I)}< by Theorem 1.1. Thus we have the following result which is a consequence of Theorem 1.2.

    Corollary 2.9. Let R be a Ding-Chen ring. Then R has finite left Gorenstein global dimension if and only if there exists a nonnegative integer n, such that

    sup{G(I)codimPPG(P)}=sup{G(P)dimIIG(I)}n.

    Let Λ be an artin algebra over a commutative artinian ring k and modΛ the category of finitely generated left Λ-modules. Suppose F is an additive subbifunctor of the additive bifunctor Ext1Λ(,):(modΛ)op×modΛAb. A short exact sequence η:0XfYgZ0 in modΛ is said to be F-exact if η is in F(Z,X). The full subcategory of modΛ consisting of all F-projective (respectively, F-injective) modules is denoted by P(F) (respectively, I(F)). F is said to have enough projectives (respectively, injectives) if for any AmodΛ there is an F-exact sequence 0BPA0 (respectively, 0AIC0) with P in P(F) (respectively, I in I(F)). Assume that F has enough projectives and injectives. Then (P(F),I(F)) is an admissible balanced pair in modΛ. The notions of F-Gorenstein projective modules and F-Gorenstein injective modules were introduced in [15]. It is not hard to see that in this case, G(P(F)) coincides with F-Gorenstein projective modules. Dually, G(I(F)) coincides with the F-Gorenstein injective modules. Denote by pdFM (respectively, idFM) the F-projective (respectively, F-injective) dimension of M. As a consequence of Theorem 1.2, we have the following result which contains [15,Theorem 3.4].

    Corollary 2.10. Let Λ be an artin algebra and F an additive subbifunctor of the additive bifunctor Ext1Λ(,). Then the following are equivalent for any nonnegative integer n:

    (1) For any finitely generated left Λ-module A, G(P(F))dimAn;

    (2) For any finitely generated left Λ-module A, G(I(F))codimAn;

    (3) sup{idFPPP(F)}=sup{pdFIII(F)}n.

    Moreover,

    sup{G(P(F))dimMMmodΛ}=sup{G(I(F))codimMMmodΛ}.

    The authors thank the referees for their careful reading and excellent suggestions.



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