Dynamic analysis of two fishery capture models with a variable search rate and fuzzy biological parameters

: The fishery resource is a kind of important renewable resource and it is closely connected with people’s production and life. However, fishery resources are not inexhaustible, so it has become an important research topic to develop fishery resources reasonably and ensure their sustainability. In the current study, considering the environment changes in the system, a fishery model with a variable predator search rate and fuzzy biological parameters was established first and then two modes of capture strategies were introduced to achieve fishery resource exploitation. For the fishery model in a continuous capture mode, the dynamic properties were analyzed and the results show that predator search rate, imprecision indexes and capture e ff orts have a certain impact on the existence and stability of the coexistence equilibrium. The bionomic equilibrium and optimal capture strategy were also discussed. For the fishery model in a state-dependent feedback capture mode, the complex dynamics including the existence and stability of the periodic solutions were investigated. Besides the theoretical results, numerical simulations were implemented step by step and the e ff ects of predator search rate, fuzzy biological parameters and capture e ff orts on the system were demonstrated. This study not only enriched the related content of fishery dynamics, but also provided certain reference for the development and utilization of fishery resources under the environment with uncertain parameters.


Introduction
In the natural ecosystem, forests, grasslands, wildlife and fishery resources are different important components, providing rich natural resources for human production and life, and playing an important role in promoting human development.Fishery resource, as a valuable renewable resource, is closely related to our daily life.The Food and Agriculture Organization of the United Nations (FAO) reported in 2018 that the total world fisheries production recorded new highs, but nearly one third of fisheries production was wasted.Meanwhile, overfishing is widespread, which poses a huge challenge to the sustainability of the world's fish supply [1].The ocean is a large ecosystem, and the marine species are complex and form both competition and predation relationships with each other.The research on predator-prey interactions and their dynamic behavior among populations has long been a topic of common interest to mathematicians and biologists.Especially in fishery, studying the rationality behind the dynamic behavior of fish species can help humans develop and utilize fishery resources more scientifically and rationally.
The essence of biomathematics is to transform a complex biological problem into a mathematical problem by means of establishing mathematical models.The pioneer work investigating the predation relationship was carried out by Lotka [2] and Volterra [3], which is widely referred to the Lotka-Volterra model.Since then, many scholars have improved the classic Lotka-Volterra model to describe various biological phenomena.For example, Gause et al. [4] introduced a Gause-type model with a general form of uptake function.Smith [5] introduced a Smith model in which a species growth is low in the presence of limited food and it is assumed that after the population reaches saturation, the population will not increase any more and the food is only used to maintain the survival of the species.Compared with the logistic growth function, the Smith growth function in some cases is more accurate in describing the growth law.Based on this consideration, Sivakumar et al. [6] introduced a diffusive model with Smith growth and analyzed the stability and hopf bifurcation.Han et al. [7] introduced a spatiotemporal discrete model with Smith growth rate function and analyzed the bifurcation and turing instability.Feng et al. [8] introduced a modified Leslie-Gower incorporated with the Smith growth rate and Beddington-DeAngelis uptake function.Another aspect of the work to improve the model is reflected in the functional reaction functions [9][10][11][12], among others the Holling-II type uptake function is widely applicable to many species, and the model with the Holling-II type uptake function has been widely concerned by scholars.In Holling-II uptake function, the search rate is often assumed to be constant.However, in the actual system, the density of prey and the search environment of predators have obvious effects on the search rate of predators.Based on the above biological background, Hassell and Comins [13] introduced a saturated search rate, and subsequently Yu et al. [14] analyzed the predator-prey model with the variable predator search rate and fear effect.In this study, we introduced a Smith growth predator-prey model with the Holling-II functional response and a variable search rate.
With the change of the external environment, such as extreme environment and temperature, the biological parameters of some species including the birth and death rates will show certain fluctuations, which will further influence the dynamics of species populations.Some findings suggest that fish species are more affected by climate change [15][16][17].Kwon et al. [15] showed that climate changes can make fish populations decrease.Although environmental factors are known to influence population change, it is not clear how these factors affect population change due to certain fluctuations and uncertainties.To describe this uncertainty, scholars introduced some specific representations in predator-prey models, such as random disturbances [18][19][20][21], fuzzy numbers and fuzzy sets [22][23][24][25][26][27][28][29][30][31][32].Pal et al. [22] introduced a predator-prey model with fuzzy interval numbers represented by interval-valued function.Pal et al. [23] investigated a fishery model with fuzzy parameters and human capture activities.Pal et al. [24] studied a delayed predator-prey harvesting model with interval-valued imprecise parameters and presented a stability and bifurcation analysis method.Yu et al. [25] analyzed a predator-prey capture model with interval-valued imprecise parameters considering the mutual interference between predators in the system as well as the prey refuge effect.Das et al. [26] considered the phenomenon of pest disease transmission in the system and studied a class of predator-prey models with interval-valued parameters.In most of the above papers and related studies, the same index is used to describe different interval-valued parameters.In order to describe different parameters, Xiao et al. [27] used interval-valued numbers with different indexes to describe imprecise parameters and studied a two-species competitive model with interval-valued fuzzy parameters.Tian et al. [28][29][30] used interval-valued fuzzy numbers with different indicators to describe the parameters of different populations, and studied two fishery capture models with parameters represented by species-dependent interval-valued functions.Yu et al. [31] investigated a fuzzy predator-prey model with parameters represented by a triangular fuzzy number.Xu et al. [32] developed a triangular fuzzy water hyacinth-fish model with a Kuznets curve effect.
Fishery resource is closely connected with people's production and life, and fishery capture activity is an indispensable link for humans to obtain natural resources, which can be divided into continuous ways [33][34][35][36][37][38] or intermittent ways [39][40][41][42][43][44][45].The continuous harvesting activity is easy to implement in practice, and easy to simulate in the modeling process.However, this kind of harvesting method is relatively ideal and does not take into account the fishery environment and the current situation of fish resources.In order to realize the rational development and utilization of fish resources and promote the healthy and sustainable development of fish resources, it is necessary to adopt more reasonable and effective ways.Compared with continuous harvesting activity, intermittent harvesting activity, which occurs at discrete moments, is more realistic in practice.Among the many types of intermittent harvesting activities, state-dependent harvesting activity is a typical one, which takes full account of the current situation of prey or predators and promotes the sustainable development of fish species as far as possible.For such kinds of human interventions, impulsive differential equations presents an effective description [46][47][48][49][50][51][52][53][54][55][56][57][58][59].In the above studies, the models involved can be roughly divided into four types: Prey-dependent [41,45,[51][52][53][54][55][56][57], predator-dependent [39,40], ratio-dependent [58] and hybrid-dependent [42][43][44].From the perspective of keeping ecological balance, setting certain threshold values for both prey populations and predator populations is necessary.In this study, we develop a switch capture strategy, where fishing activities can be carried out only if prey population reaches or exceeds a preset threshold.At the same time, if predator populations are below a low value, we will not only carry out fishing activities but also release some predator pups.
Inspired by the above work, in the current study we propose two fishing models with variable predator search speed and fuzzy parameters and discuss the effects of different predator search rates and parameter imprecision indicators, as well as the dynamic behavior guided by different fishing methods.We organize the article in the following way: In the subsequent section, we develop two fishery capture models with variable search rate and triangular fuzzy biological parameters, followed by a presentation of basic concepts of impulsive semi-continuous dynamic system.In section three, we mainly investigate the system dynamics under continuous capture strategy and complex dynamic behaviour induced by the switched capture strategy.In the next section, we conduct numerical simulations to illustrate the conclusions of the previous section.At the end of the article, we make a brief summary of the research.

Mathematical model and basic knowledge
In this work, we consider a Gause-type fishery model with Smith growth and Holling-II type uptake function, i.e., in which x and y indicate the density of prey and predator species, respectively; parameters r and K are intrinsic growth rate and maximum environmental capacity, respectively; the term rx(K − x)/(K + mx) is the Smith growth function with maintenance constant m; H denotes the handling time of a prey by one predator; d is the predator's death rate; c denotes the conversion coefficient.The parameter s describes the search rate of the predator and is generally taken as a constant.However, in a real system, the predator's search speed will depend on the density of the prey and the search environment.Considering this phenomenon, Hassell and Comins [13] introduced the following type of search rate where a is the maximum search rate and g is a constant, then the extended form of Model (2.1) is described as follows: (2.2) In natural systems, biological species are inevitably affected by environmental changes, so it is meaningful and necessary to consider biological models with imprecise parameters (2.2).In order to describe the uncertainty of such parameters, we use triangular fuzzy number (TFN) [31].For a triangular fuzzy number Considering that the mortality and conversion rates of predators and prey are most susceptible to environmental changes, imprecisions of these three parameters are assumed in this paper, and represented by triangular fuzzy number r = (r 1 , r 2 , r 3 ), d = (d 1 , d 2 , d 3 ) and c = (c 1 , c 2 , c 3 ).Using theory of ω-cut fuzzy number, Model (2.2) can be expressed as (2.3) Using the utility function method (UFM) [23,32], one can get where 0 ≤ w 1 ≤ 1, 0 ≤ w 2 ≤ 1.
For convenience, let's write: r = w 1 r l(ω) . By combining formulas (2.3) and (2.4), we get (2.5) In order to meet people's daily life, it is necessary to capture two kinds of fish.Let q 1 and E 1 be the capture rate and capture effort of the prey species, q 2 and E 2 be that of the predators.To analyze the impact of different fishing strategies on the system, we consider two different forms of fishing strategies.The first capture strategy is a continuous mode, and the fishery model based on the continuous capture mode is formulated as follows: The second is a state-dependent feedback capture strategy.Let x T be the prey's reference threshold.When the prey populations reach the threshold x T , a capture activity is implemented.In addition, to maintain the balance of the ecosystem, it is also necessary to consider whether to release predator fish.Let y T be the predator's reference threshold.When the predator population is below the threshold y T , in addition to capture activity, a certain quantity of predators needs to be put into the system, which is denoted by τ.Then, the fishery model based on the switch capture strategy is formulated as follows: (2.7) The objective is to investigate the effects of variable search rate and imprecise biological parameters on Model (2.5), while exploring the complex dynamic behavior of Models (2.6) and (2.7) under different capture modes.
In Figure 1, we can define the poincaré map on N by P M ≜ I • π, i.e., L 4 = P M (L 2 ) = P M (L 1 ).Then two types of successor functions are defined: Definition 2.6 (Order-k periodic solution [28,42,43]).For a given is orbitally asymptotically stable, where

Main results
For convenience of description, let us denote E = (E 1 , E 2 ) and define 3.1.Dynamic characteristics of Model (2.6) 3.1.1.Existence and stability of equilibrium Theorem 3.1.For Model (2.6), an extinction equilibrium Q 0 (0, 0) always exists; Q 1 (K E , 0) exists as long as The existence of trivial and predator extinction equilibria is evident.The coexistent equilibrium should satisfy the following equations: From Eq (3.1), a quadratic equation can be obtained: where It can be concluded that Eq (3. 3) has a positive root is locally asymptotically stable, where For Q 0 , there is The eigenvalues are λ 1 = rK−q 1 E 1 and λ 2 = − d−q 2 E 2 < 0. When E 1 > E 1 , there is λ 1 = r−q 1 E 1 < 0, hence Q 0 (0, 0) is locally asymptotically stable.Moreover, if E 1 > E 1 , then dx/dt < 0, i.e., prey populations decrease to zero, which causes the number of predators to decrease to zero as well.Therefore, Q 0 (0, 0) is globally asymptotically stable.
For Q 1 , there is The eigenvalues are Combine cases i) and ii) with the existence condition of K E , i.e., 0 ≤ E 1 < E 1 , and it can be derived that λ 1 < 0 and λ 2 < 0, thus, Q 1 (K (E 1 ,E 2 ) ), 0) is locally asymptotically stable.
For Q 2 , there is where Economic benefit is important for human activities; thus, it is meaningful to combine biological balance with economic benefit balance.Let s 1 and s 2 be the selling price of prey and predator, c 1 and c 2 be the costs per unit od capture effort accordingly, then the net profit of the capture process can be characterized as The bionomic equilibrium (x r , x r , E 1r , E 2r ) satisfies that ) To determine a bionomic equilibrium (x r , y r , E 1r , E 2r ), we will discuss it in four cases: Case 1: c 2 > s 2 q 2 y.In this case, the predator's fishing costs outweigh its benefits, which means that E 2r = 0. Thus, only the prey stocks are fished and (c 1 < s 1 q 1 x), then x r = c 1 /s 1 q 1 and (y r , E 1r ) satisfy Case 2: c 1 > s 1 q 1 x.In this case, the prey's fishing costs outweigh its benefits, which means that E 1r = 0. Thus, only the prey stocks are fished and (c 2 < s 2 q 2 y), then we have y r = c 2 /s 2 q 2 .Substituting y r , E 1r for y, E 1 respectively in Eq (3.3) yields that If there exists positive solution of Eq (3.7), denoted as x r , then we have Thus, E 2r > 0 in the case of d < ĉax r 2 /(aHx r 2 + x r + g).
Case 3: If c 1 > s 1 q 1 x and c 2 > s 2 q 2 y, fishery activities for both prey and predator will be quit.
Case 4: If c 1 < s 1 q 1 x and c 2 < s 2 q 2 y, there are x r = c 1 /s 1 q 1 and y r = c 2 /s 2 q 2 .Substituting into Eqs (3.3) and (3.4), one can get Therefore, and Therefore, it can be concluded that the nontrivial bionomic equilibrium point (x r , y r , E 1r , E 2r ) exists if inequations (3.8) and (3.9) hold.

Optimal harvesting strategy
Let δ be the discount rate, then the net profit J (E) is defined as where E 1 min ≤ E 1 (t) ≤ E 1 max and E 2 min ≤ E 2 (t) ≤ E 2 max .According to Pontryagin's maximal principle [60], denote in which λ 1 and λ 2 are to be determined.clearly, H is linear dependent on E 1 and E 2 .We suppose that the control are not bangbang ones, which means the optimal harvesting efforts would not be E i min or E i max , and the singular control is obtained by The adjoint equations are dλ 2 dt = − ∂H ∂y = − e −δt q 2 s 2 E 2 + λ 1 F x (x, y; w 1 , w 2 , ω, E 1 ) + λ 2 G y (x, y; w 1 , w 2 , ω, E 2 ) . (3.12) Substituting λ 1 and λ 2 into Eq (3.11) and simplifying, there is (aHx 2 +x+g) 2 y .

Complex dynamics of Model (2.7)
Define

Simulations for Model (2.6)
We will illustrate the effects of the imprecision indicators w 1 , w 2 , ω, search rate constant g and capture efforts E = (E 1 , E 2 ) on the system, respectively.
First, we illustrate the impact of the imprecision indicators w 1 , w 2 , ω on Model (2.6).For g = 0.05, E = 0 and (x 0 , y 0 ) = (10, 4), the time series and stable steady states for different w 1 , w 2 , ω are presented in Figures 3-5 and Table 1.when ω = 0.5, where the density of prey is marked blue and the density of predators is marked red.when ω = 0.9, where the density of prey is marked blue and the density of predators is marked red.From Figures 3-5 and Table 1 it can be concluded that: 1) When w 1 is increasing, x * E doesn't change, while y * E decreases; 2) when w 2 is increasing, x * E is increasing, while y * E is decreasing; 3) when ω is increasing, x * E is also increasing, while y * E is decreasing; 4) as w 2 decreases to a certain level, the coexistence equilibrium Q 2 disappears and the predator species goes extinct.
Second, we illustrate the impact of predator search rate on Model (2.6).Here, the parameter g is selected as the relative size of predator search rate.For w 1 = 0.2, w 2 = 0.2, ω = 0.5, (E 1 , E 2 ) = (0.015, 0.001) and (x 0 , y 0 ) = (10, 4), the impact of g on Model (2.6) is presented in Figure 6 and Table 2.It can be observed that the predator search rate has a certain impact on the stability of the coexistence equilibrium Q 2 .As g increases, i.e., predator search speed decreases, x * E increases and y * E decreases.Especially, when g increases to a certain level, the predator search rate becomes extremely small and leads to the extinction of predator species.
Third, we illustrate the impact of (E 1 , E 2 ) on Model (2.6).For g = 0.05, w 1 = 0.2, w 2 = 0.2, ω = 0.5, the impact of (E 1 , E 2 ) on the system with the initial value (x 0 = 10, y 0 = 4) is presented in Figure 7.It can be observed that the fishing activities contribute to the stability of the system.Moreover, as long as the fishing intensity is appropriate, the system can achieve a coexistence steady state.However, if the fishing effort is too large, especially if the capture effort of the prey population exceeds the given threshold, the system will become extinct.These observations and the conclusion are consistent with Theorem 1.
Next, we illustrate the impact of the imprecision indicators w 1 , w 2 , ω on bionomic equilibrium.For s 1 = 0.5, s 2 = 20, c 1 = 1.8, c 1 = 2, the impact of w 1 , w 2 , ω on bionomic equilibrium (if existing) is presented in Table 3.It can be observed that bionomic equilibrium does not exist unconditionally and needs to satisfy inequalities (3.8) and (3.9).

Conclusions
Considering that some biological parameters of different species in an ecosystem fluctuate to a certain extent due to changes in the external environment, it is of practical significance to study biological models with imprecise parameters.In addition, the search rate of predators varies rather than is fixed in response to changes in the environment and the distribution of prey.In view of the above phenomena, we proposed a Gause-type fishery model incorporated with the Smith growth function, variable search rate and triangular fuzzy biological parameters.Moreover, from the perspective of rational exploitation of fishery resources, we introduced two fishing strategies into the system and analyzed the effects of different fishing strategies on fish resources.
For the continuous type capture system, we figured out the effects of variable predator search speed and imprecision indicators on the system's dynamics (Theorems 1-2, Tables 1,2 and Figures 3-6).The results show that when the parameters change obviously, the imprecise indicators have a certain influence on the dynamic characteristics of the system.This is reasonable since the gradual change of the environment will not change the related characteristics of biological species, but dramatic changes will change its living habits to adapt to this change.In addition, for the sake of maximizing economic benefit of the capture process, we discussed the bionomic equilibrium of the system (Table 3) and the optimal capture strategy.
For the switch capture system, we provided the existence and stability conditions of predatorextinction periodic solution and coexistence order-1 or 2 periodic solution.To prevent predators from going extinct, predator populations should not be captured too aggressively.Through computer simulations, we found different order coexistence periodic solutions.This further indicates that the ecological balance of species can be achieved with the switch capture strategy.
The research indicated that the triangular imprecise biological parameters, variable predator search rate and capture activities further enriched the dynamic characteristics of the biological system, and also provided a theoretical reference basis for scientific and effective exploitation and utilization of fishery resources under uncertain parameter environments.

Figure 1 .
Figure 1.Schematic representation of the successor points.

9 Figure 3 .
Figure 3. Impact of imprecision indicators on Model (2.6):Time series for different (w 1 , w 2 )when ω = 0.1, where the density of prey is marked blue and the density of predators is marked red.

Figure 4 .
Figure 4. Impact of imprecision indicators on Model (2.6):Time series for different (w 1 , w 2 )when ω = 0.5, where the density of prey is marked blue and the density of predators is marked red.

5 Figure 5 .
Figure 5. Impact of imprecision indicators on Model (2.6):Time series for different (w 1 , w 2 )when ω = 0.9, where the density of prey is marked blue and the density of predators is marked red.

Figure 6 .
Figure 6.Impact of predator search speed g on the Model (2.6): the density of prey is marked blue and the density of predators is marked red.

Figure 7 .
Figure 7. Impact of capture effects (E 1 , E 2 ) on Model (2.6): the density of prey is marked blue and the density of predators is marked red.
3, E 1 = 5, E 2 = 10, Model (2.7) has a different coexistence periodic solution, as shown in Figures 9-12.It can be observed that the periodic solution depends on the value of τ.

Figure 8 .Figure 9 .
Figure 8.Time series and phase diagram of the predator-extinction periodic solution in case of τ = 0.

Figure 10 .
Figure 10.Time series and phase diagram of the coexistence order-2 periodic solution in case of τ = 1.2.

Figure 11 .
Figure 11.Time series and phase diagram of the coexistence order-3 periodic solution in case of τ = 1.

Figure 12 .
Figure 12.Time series and phase diagram of the coexistence order-9 periodic solution in case of τ = 0.7.

Table 2 .
The impact of predator search speed on coexistence equilibrium Q 2 .