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
Let
Let
Theorem 1.1. (see [4,Theorem 4.1]) Let
(1) For any
(2) For any
(3)
Moreover,
sup{G(P)−dimM∣M∈ModR}=sup{G(I)−codimM∣M∈ModR}. |
In this case, we say that
The main goal of this paper is to generalize Theorem 1.1 to Gorenstein subcategories
Theorem 1.2. Let
(1) For any object
(2) For any object
(3)
Moreover,
sup{G(X)−dimM∣M∈A}=sup{G(Y)−codimM∣M∈A}. |
The common value of the last equality is called the Gorenstein global dimension of the abelian category
The proof of the above results will be carried out in the next section.
Throughout this section, we always assume that
Definition 2.1. (see [3,Definition 1.1]) A pair
(BP0) the subcategory
(BP1) for each object
(BP2) for each object
We say that a contravariantly finite subcategory
Lemma 2.2. If the short exact sequence
(1) If
(2) If
(3) If
(4) If
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
(2) The "only if" part is clear. For the "if" part, since
sup{G(X)−dimM∣M∈A}=sup{G(Y)−codimM∣M∈A}. |
By [3,Propsotion 2.2], all rows and columns are
Recall that the
sup{G(X)−dimM∣M∈A}=sup{G(Y)−codimM∣M∈A}. |
If there is no such an integer, set
Lemma 2.3. The following are true for any object
(1) If
(2) If
Proof. We just prove (1) since (2) follows by duality. If
0→Pn→Pn−1→⋯→P1→P0→A→0 |
with all
0→K−j→Q−j−1→K−j−1→0 |
where
We let
Proposition 2.4. The following are true for any
(1) If
In this case,
(2) If
In this case,
Proof. We just prove (1) since (2) follows by duality.
Proposition 2.5. The following are true for any
(1) If
0→K→G→A→0 and 0→A→L→G′→0 |
such that
(2) If
0→A→G→K→0 and 0→G′→L→A→0 |
such that
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
(1) If
(2) If
Proof. We just prove (1) since (2) follows by duality. It is clear
0→A→G→K→0 and 0→G′→L→A→0 |
where all rows and columns are
Proposition 2.7. If
sup{Y−cores.dimP∣P∈X}=sup{X−res.dimI∣I∈Y}⩽n. |
Proof. Suppose
Let
Proof of Theorem 1.2.
Let
0→A→I0→I1→⋯ |
is a
0→Ki→Ii→Ki+1→0,i⩾0. |
Here
0→Ki→Ii→Ki+1→0,i⩾0. |
with
Ωn(Ki)→Ωn(Ii)→Ωn(Ki+1),i⩾0, |
together, there is a
0→Ωn(A)→Ωn(I0)→Ωn(I1)→⋯ |
with
⋯→Ωn(P1)→Ωn(P0)→Ωn(A)→0 |
with
⋯→Ωn(P1)→Ωn(P0)→Ωn(I0)→Ωn(I1)→⋯ |
with each term in
Dually, one can prove
The last equality is immediate from above equivalences and Proposition 2.7.
Let
Corollary 2.8. Let
sup{pidRP∣P∈PP(R)}=sup{ppdRI∣I∈PI(R)}⩽n. |
Let
Corollary 2.9. Let
sup{G(I)−codimP∣P∈G(P)}=sup{G(P)−dimI∣I∈G(I)}⩽n. |
Let
Corollary 2.10. Let
(1) For any finitely generated left
(2) For any finitely generated left
(3)
Moreover,
sup{G(P(F))−dimM∣M∈modΛ}=sup{G(I(F))−codimM∣M∈modΛ}. |
The authors thank the referees for their careful reading and excellent suggestions.
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