Abstract
We explore existence of fixed points, topological classifications around fixed points, existence of periodic points and prime period, and bifurcation analysis of a three-species discrete food chain model with harvesting. Finally, theoretical results are numerically verified.
1. Introduction
Many different types of interactions exist in nature between various species of organisms on this planet Earth and are studied under the discipline of ecology. Ecological interactions are most fundamental part in biology which determines community structure and development. Not all interactions are positive, some are negative also. One of the examples of negative correlation is ammensalism. Ammensalism is a type of ecological interactions between the members of two different species in which one is harmed, destroyed, or inhabited by the member of another species, while the other remains unaffected, neither harmed nor benefitted. It is a type of competitive behavior among different species and is frequently used to refer to asymmetrical competitive association. Research in the field of ecology draws the attention of several mathematicians such as Lotka [1] and Volterra [2]. Nowadays, ecologist and mathematician jointly contributed to the growth of this area of knowledge. Recently, many researchers have investigated the dynamical properties of discrete-time ecological models such as prey-predation, competitions, neutralism, and mutualism by studying fixed points, local and global attractivity, bounded, existence of bifurcation, and many more. For instance, Beddington et al. [3] have explored the behavior of following predator-prey model:
Chen [4] has explored global attractivity and permanence of the following discrete multispecies system:
Fang and Chen [5] have explored the permanence of multispecies Lotka–Volterra competition predator-prey system with delays. Furthermore, Fang et al. [6] have explored the dynamics of the following system:
Agiza et al. [7] have explored chaotic dynamics of the following discrete model with Holling type II:
Huo and Li [8] have explored stable periodic solution of the following discrete model:
Lu and Zhang [9] have studied the permanence and global attractivity of the following discrete system:
Zhao and Zhang [10] explored the chaos and permanence of the following discrete model:
Zhao et al. [11] have investigated the dynamics of the following discrete model:where .
On the contrary, in recent years, many papers have been published to investigate the bifurcation analysis of certain discrete models by choosing step size as a bifurcation parameter. For example, Salman et al. [12] have explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Liu and Xiao [13] have explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Hasan and Hama [14] have explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Wu and Zhang [15] have explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Rana [16] has explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Rana and Kulsum [17] have explored bifurcation analysis of the following discrete system:by choosing step size as a bifurcation parameter. Motivated from the aforementioned studies, in this work, we explore existence of fixed points, topological classifications around fixed points, periodic points, and bifurcation analysis, by choosing step size as a bifurcation parameter, of the following three species discrete food chain model with harvesting:which is a discrete form of the following model:by Euler forward formula, where is step size and is customary denoted by . It is noted that, in model (16), , , and , respectively, denote populations of first, second, and third species. Moreover denotes growth rate of first, second, and third species; denotes the rate of decrease of first, second, and third species due to internal competitions; denotes rate of decrease of second species due to attack of first species; denotes the rate of decrease of third species due to attack of second species; and , respectively, denote the harvesting rate of first and second species. It is also important to note that all parameters , and are positive. In addition, it is important here to mention that we will explore dynamical properties of the discrete-time model (15) instead of the continuous-time model, which is depicted in (16), because discrete-time models governed by difference equations are more realistic and appropriate than the continuous ones in the case where populations have nonoverlapping generations, and moreover, discrete models can also provide efficient computational results for numerical simulations [12, 13].
This paper is structured as follows. In Section 2, we study the existence of fixed points of model (15) algebraically. The linearized form of model (15) is presented in Section 3. In Section 4, we explored topological classification around fixed points of the model. Existence of periodic points of model (15) is explored in Section 5. In Section 6, we explored detailed analysis of bifurcation around fixed points of model (15). Theoretical results are verified numerically in Section 7. Brief summary of the paper is presented in Section 8.
2. Study of Equilibrium Points
Here, we will study the boundary and interior equilibrium points of model (15) as follows.
Lemma 1. Model (15) has atmost eight equilibrium points in . Precisely,(i); model (15) has a trivial equilibrium point: (ii); model (15) has boundary equilibrium point: (iii) is a boundary equilibrium point of (15) if (iv) is a boundary equilibrium point of (15) if (v) is a boundary equilibrium point of (15) if with (vi) is a boundary equilibrium point of (15) if (vii) is a boundary equilibrium point of (15) if with (viii) is an interior equilibrium point of (15) if , and
Proof. If model (15) has an equilibrium point, , thenThe simple computation yields that, for the values of , (17) satisfied identically. So, one can conclude that model (15) has seven boundary points: . In order to find interior point, from (17), one obtainsFrom of (18), one obtainsFrom equation of (18) and (19), one obtainsFrom equation of (18) and (20), one obtainsFrom (19)–(21), one can conclude that is an interior equilibrium point of (15) if , , and .
3. Linearized Form of Model (15)
The variational matrix about under the map:whereiswith
4. Dynamical Behavior: Topological Properties of Equilibrium Points
The dynamical behavior about fixed points of model (15) is explored in this section.
4.1. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about of model (15) is concluded as follows.
Lemma 2. (i)For all allowed parametric values, , is not sink.(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic ifor
4.2. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows:
Lemma 3. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
4.3. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows.
Lemma 4. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
4.4. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows.
Lemma 5. (i)For all allowed parametric values, , is not sink.(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic ifor
4.5. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows.
Lemma 6. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
4.6. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows.
Lemma 7. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
4.7. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows:
Lemma 8. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
4.8. Dynamical Behavior about
From (25), eigenvalues of about are
The dynamical behavior about is concluded as follows.
Lemma 9. (i) is a sink if(ii) is a source if(iii) is a saddle if(iv) is nonhyperbolic iforor
5. Periodic Points
We will prove that of model (15) are periodic points of period .
Theorem 1. Equilibrium points of model (15) are periodic points of prime period 1.
Proof. From (15), definewhere , and are represented in (23). From (78), the computation yieldsHence, from (79), we can say that equilibrium points of three species model (15) are periodic points of prime period 1.
Now, it is proved that equilibrium points are period points of period .
Theorem 2. of model (15) is a periodic point of period .
Proof. From (78), the following computation yields the required statement:
Theorem 3. of model (15) is a periodic point of period .
Proof. Utilizing the computation as we have done in (80), one gets the following required statement:
Theorem 4. of model (15) is a periodic point of period .
Proof. In view of (80), one gets the following required statement:
Theorem 5. of model (15) is a periodic point of period .
Proof. In view of (80), one gets the following required statement:
Theorem 6. of model (15) is a periodic point of period .
Proof. From (80), one obtains
Theorem 7. of model (15) is a periodic point of period .
Proof. From (80), one obtains
Theorem 8. of model (15) is a periodic point of period .
Proof. From (80), one obtains
Theorem 9. of model (15) is a periodic point of period .
Proof. From (80), one obtains
6. Analysis of Bifurcation
In this section, we give analysis of bifurcation about fixed points of model (15) by bifurcation theory [18, 19].
6.1. Analysis of Bifurcation at
Here, we will study analysis of bifurcation at of model (15). From (26), the simple computation yields , but . This suggests that model (15) could undergo a flip bifurcation around if passes the curve:
However, flip bifurcation cannot occur by computation, so is degenerated with high co-dimension as .
6.2. Analysis of Bifurcation at
We will study analysis of bifurcation at of model (15). From (26), the simple computation yields , but . This suggests that model (15) could undergo a flip bifurcation around if passes the curve:
The proof of following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 10. Model (15) undergo flip bifurcation around if .
Proof. It is noticed that three-species model (15) is invariant with respect to . Thus, we restrict (15) on , to determine the bifurcation, where it takes the formFrom (90), defineNow, one denotes . The computation yieldsFrom (92)–(94), it can be concluded that the model undergoes flip bifurcation around if .
6.3. Analysis of Bifurcation at
From (38), the computation yields , but . This suggests that model (15) could undergo a flip bifurcation around if passes the curve:
The proof of following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 11. Model (15) undergoes flip bifurcation around if .
Proof. It is noticed that, , model (15) is invariant. So, one restricts model (15) on , where it becomesFrom (96), defineDenote . By computation, one obtainsSo, model (15) undergoes flip bifurcation by (98)–(100) if .
6.4. Analysis of Bifurcation at
From (45), the computation yields , but . This suggests that model (15) could undergo a flip bifurcation around if passes the curve:
The proof of the following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 12. Model (15) undergoes flip bifurcation around if .
Proof. It is noticed that, , model (15) is invariant. So, one restricts model (15) on , where it becomesFrom (102), defineDenote . By computation, one obtainsSo, model (15) undergoes flip bifurcation by (104)–(106) if .
6.5. Analysis of Bifurcation at
From (50), the computation yields , but . This suggests that model (15) could undergo flip bifurcation around if passes the curve:
The proof of the following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 13. Model (15) undergoes flip bifurcation around if .
Proof. Recall that if , then , but . So, hereafter, detailed flip bifurcation is explored if varies in the nbhd of , , by assuming . LetThen, (15) givesBy using transformation,(109) takes the formwhereNow, consider (111) on the center manifold, .,whereFrom (111) and (113), one obtainsFrom (115), computation yields . Finally, map (111); restrict to asFor the model to undergo flip bifurcation, the following should be nonzero:From (117), one can say that about model (15) undergoes flip bifurcation if . Moreover, period-2 points bifurcating from are stable since .
6.6. Analysis of Bifurcation at
From (57), the computation yields , but . This suggests that model (15) could undergo flip bifurcation around if passes the curve:
The proof of the following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 14. Model (15) undergoes flip bifurcation around if .
Proof. Recall that if , then , but . So, hereafter, detailed flip bifurcation is explored if varies in the nbhd of , , by assuming . LetThen, (15) givesUsing transformation,(120) becomeswhereNow, from model (122) on the center manifold,whereFrom (122) and (124), one hasFrom (126), the calculation yields: . Thus, map (122); restrict to asFrom (117) and (127), the computation yields: and . This implies that about model (15) undergoes flip bifurcation if . Moreover, period-2 points bifurcating from are stable since .
6.7. Analysis of Bifurcation at
From (64), the computation yields , but , . This suggests that model (15) could undergo flip bifurcation around if passes the curve:
The proof of the following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 15. Model (15) undergoes flip bifurcation around if .
Proof. Recall that if , then , but , . So, in the following, flip bifurcation is explored by assuming . LetThen, (15) givesUsing transformation,(130) giveswhereNow, using system (132) on the center manifold,whereIn view of (132) and (134), we obtainFrom (136), one gets: . Finally, map (132); restrict to asFrom (117) and (137), the computation yields: . Moreover, . This implies that about model (15) undergoes flip bifurcation if . Moreover, period-2 points bifurcating from are stable since .
6.8. Analysis of Bifurcation at
From (71), the computation yields , but , . This suggests that model (15) could undergo flip bifurcation around if passes the curve:
The proof of the following theorem shows that model (15) undergoes flip bifurcation around if .
Theorem 16. Model (15) undergoes flip bifurcation around if .
Proof. Recall that if , then , but , . So, in the following, flip bifurcation is explored by assuming . LetThen, (15) becomeswhereNow, by utilizing transformation,giveswhereNow, from system (143) on the center manifoldwhereFrom (143) and (145), we obtainFrom (147), computation yields: . Finally, map (143); restrict to asFrom (117) and (148), the computation yields: . Moreover, . This implies that about model (15) undergoes flip bifurcation if . Moreover, period-2 points bifurcating from are stable since .
7. Numerical Simulations
Numerical simulations of three-species model (15) are performed in this section to check previous theoretical findings and to show rich dynamical behaviors. In this regard, following eight cases are presented to address the accuracy of theoretical results obtained about fixed points for model (15): Case I: if , then, from (29), one gets . From (27), if and starting from , then Figure 1(a) indicates that of (15) is a source. However, if , then Figure 1(b) indicates that of (15) is a saddle. Hence, theoretical results obtained in Lemma 2 coincide with numerical simulations. Case II: if , then, from (35), one gets . Figure 2(a) indicates if , then of (15) is a sink. However, if , then Figure 2(b) indicates that is unstable. Moreover, if , then exchanges the stability, and in fact, flip bifurcation takes place by Theorem 10. Therefore, the flip bifurcation diagrams are presented in Figure 3. Finally, maximum Lyapunov exponents corresponding to Figure 3 are drawn in Figure 4. Case III: if , then, from (42), one gets . Hence, is stable if , and exchange stability is , and in fact, flip bifurcation takes place by Theorem 11. Therefore, the flip bifurcation diagrams with initial value are presented in Figure 5. Finally, maximum Lyapunov exponents corresponding to Figure 5 are drawn in Figure 6. Case IV: if , then, from (48), one gets . Hence, is stable if , and exchange stability is , and in fact, flip bifurcation takes place by Theorem 12. Therefore, the flip bifurcation diagrams with initial value are presented in Figure 7. Finally, maximum Lyapunov exponents corresponding to Figure 7 are drawn in Figure 8. Case V: if , then, from (54), one gets . Hence, is stable if , and exchange stability is , and in fact, flip bifurcation takes place by Theorem 13. Therefore, the flip bifurcation diagrams with initial value are presented in Figure 9. Finally, maximum Lyapunov exponents corresponding to Figure 9 are drawn in Figure 10. Case VI: if then from (61) one gets: . Hence is stable if , and exchange stability if and infact flip bifurcation takes place by Theorem 14. Therefore the flip bifurcation diagrams with initial value are presented in Figure 11. Finally maximum lypunov exponents corresponding to Figure 11 are drawn in Figure 12. Case VII: If , then, from (68), one gets . Hence, is stable if , and exchange stability is , and in fact, flip bifurcation takes place by Theorem 15. Therefore, the flip bifurcation diagrams with initial value are presented in Figure 13. Finally, maximum Lyapunov exponents corresponding to Figure 13 are drawn in Figure 14. Case VIII: if , then, from (75), one gets . Hence, is stable if , and exchange stability is , and in fact, flip bifurcation takes place by Theorem 16. Therefore, flip bifurcation diagrams with are presented in Figure 15 which indicates that period-2 points bifurcate from are stable, since . Finally, maximum Lyapunov exponents corresponding to Figure 15 are drawn in Figure 16.
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8. Conclusion
The work is about the existence of fixed points, topological classifications around fixed points, periodic points, and bifurcations of a three-species discrete food chain model with harvesting in the region: . We proved that, for all parametric values , , , , , , , , , , and , model (15) has trivial fixed point: ; boundary fixed points: ; ; ; ; ; . We also proved that if , and ; then, is an interior equilibrium point of (15). Furthermore, we studied the local stability with different topological classifications around each fixed points whose main findings are presented in Table 1. Next, for under consideration model (15), we also studied existence of periodic points by existing theory. Furthermore, we explored the existence of possible bifurcations about each fixed points in order to understand dynamics of model (15) deeply. It is proved that (i) around model undergoes no flip bifurcation if , (ii) around model undergoes flip bifurcation if , (iii) around model undergoes flip bifurcation if , (iv) around model undergoes flip bifurcation if , (v) around model undergoes flip bifurcation if , (vi) around model undergoes flip bifurcation if , (vii) around model undergoes flip bifurcation if , and (viii) around model undergoes flip bifurcation if . Finally, obtained results are verified numerically. This research can provide a framework for theoretical basis and help for the research in different aspects of biology specifically in the field of ecology.
Data Availability
All the data used in this study are included within the article and the sources from where they were adopted are cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.