Numerical Analysis of the Impact of Decreasing the Intra-specific Competition of the Second Plant Specie on Biodiversity Scenario

doi:10.21203/rs.3.rs-3616050/v1

The intra-specific coefficients are used in a scientific method that goes beyond creating a biodiversity index to provide a numerical evaluation of various biodiversity scenarios. The ODE 45 is an efficient computational method that is used to obtain this quantification. According to our observations, biodiversity typically increases when the intra-specific coefficient is changed by percentages of 40%, 45%, 90%, 99%, and 99.99%. But the biodiversity advantage we saw at first turns into a biodiversity loss as the variances hit 101% and 105%. This shift takes place between the 99.99% and the 101% variation of the same parameter, which is the range of intra-specific coefficient variation. By using the distinct bifurcation intervals, we have found in this work, we can more effectively monitor the main worry regarding the loss of biodiversity gain, which can be interpreted as a decline in yields. Researchers studying the effects of climate change on plant species yields and related agricultural crops, such as environmentalists, numerical mathematicians, and numerical simulators, are expected to find significant value in the research findings of this study. This study presents and quantitatively discusses novel results that have not been previously reported elsewhere.

Biodiversity

Intra-specific Coefficients

Numerical

Bifurcation

ODE 45

The conventional ecological framework used to describe interactions among plant species is rooted in the notion of competing for scarce resources, as one would expect (Tilman 1998; Ford et al. 2010). This field has accumulated a substantial body of knowledge, and there exist mathematical models capable of elucidating the observed interactions in these studies (Gates et al. 1986; Gates et al. 1988; Maller 1990). Numerous researchers have shown significant interest in investigating the repercussions of extreme climate change on the interactions among plant species (Ekaka-a 2009; Ford et al. 2010; Thomas and Lee 2006; Tim 2006; Sala et al. 2002; Dauraiappah 2006; Chapin et al. 2000; Clarke and Gaston 2006; Walter et al. 2002). In our first paper on this research area (Udoh and Ekakaa-a 2021), we examined the variational impact of the intraspecific interaction coefficient within the first plant species as it interacts with itself. However, our focus in this study is specifically directed toward examining the variational impact of the intraspecific interaction coefficient within the second plant species, as it interacts with itself. We aim to draw inferences regarding its potential effects on biodiversity as compared to the results obtained in our previous work. To quantify these impacts, the current research employs a numerical modeling technique.

The complex interaction between two plant species over time (t) in days can be effectively represented by a system of first-order nonlinear ordinary differential equations, often referred to as the Lotka-Volterra system:

For the first plant species (biomass denoted as x):

$$\frac{dx}{dt}=x\left({\alpha }_{1}-{\beta }_{1}x-{\gamma }_{1}y\right)$$

1

For the second plant species (biomass denoted as y):

$$\frac{dy}{dt}=y\left({\alpha }_{2}-{\beta }_{2}y-{\gamma }_{2}x\right)$$

2

Here, the intrinsic growth rates of the x and y populations in the absence of interaction are represented by the model parameters α₁ and α₂, respectively. The parameters β₁ and β₂ denote the intra-specific or self-interaction coefficients, which act to inhibit exponential population growth. The values of γ₁ and γ₂ define the inter-specific coefficients, where γ₁ represents the impact of the second plant species y on restraining the growth of the first plant species x, and γ₂ represents the effect of the first plant species x on limiting the growth of the second plant species y.

To assess the impact of decreasing intra-specific coefficients on the biodiversity scenario, we employed the widely recognized MATLAB ODE45 function. Consistent with Udoh and Ekaka-a (2021), the subsequent parameter values will be utilized in this simulation analysis:

${\alpha }_{1}$ = 0.168, ${\beta }_{1}$ = 0.002039, ${\gamma }_{1}$ = 0.0005, ${\alpha }_{2}$ = 0.002, ${\beta }_{2}$ = 0.000015 and ${\gamma }_{2}$ = 0.00002.

The following are the complete results acquired using the above procedure as

displayed in Tables 1–7:

Table I: Calculating the effect of a 40% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BL$	$y$	${y}_{1}$	$BG$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531403718	0.000000014087827	0.250498196636003	0.250498760818603	0.000225224215891
0.349325312001738	0.349325311804536	0.000000056452292	0.250997135452566	0.250998267191912	0.000450897315611
0.412858102914633	0.412858102389365	0.000000127227354	0.251496771788616	0.251498474471933	0.000677019949294
0.487877794992095	0.487877793886822	0.000000226547075	0.251997052439703	0.251999329466729	0.000903592722414
0.576432659704310	0.576432657660767	0.000000354515498	0.252497914287299	0.252500769069596	0.001130616189292
0.680928206687639	0.680928203206613	0.000000511217856	0.252999282395326	0.253002718355410	0.001358090841857
0.804180124924089	0.804180119321288	0.000000696709763	0.253501068081790	0.253505088652076	0.001586017099109
0.949482624793687	0.949482616143982	0.000000910991307	0.254003166434270	0.254007775055768	0.001814395294056
1.120678677500689	1.120678664567710	0.000001154030921	0.254505453598243	0.254510653718968	0.002043225656139
1.322241270688331	1.322241251837262	0.000001425690573	0.255007783602803	0.255013578675843	0.002272508296809
1.559362384105503	1.559362357194244	0.000001725786111	0.255509984618200	0.255516378099369	0.002502243181954
1.838043076134275	1.838043038381554	0.000002053962820	0.256011854994581	0.256018850339608	0.002732430116067
2.165197443921075	2.165197391744548	0.000002409781505	0.256513158255958	0.256520758917065	0.002963068701582
2.548740309797457	2.548740238622494	0.000002792554558	0.257013618311200	0.257021827733075	0.003194158320703
2.997691791894467	2.997691695925222	0.000003201438058	0.257512913057984	0.257521734672900	0.003425698079334
3.522238042881883	3.522237914839923	0.000003635244357	0.258010668878118	0.258020106100257	0.003657686784818
4.133803573914338	4.133803404737609	0.000004092519773	0.258506452996558	0.258516509215214	0.003890122872586
4.845038376786851	4.845038155303167	0.000004571350472	0.258999767706050	0.259010446277811	0.004123004377843
5.669794856541051	5.669794569116058	0.000005069407279	0.259490042275745	0.259501346515312	0.004356328847041

Table II: Calculating the effect of a 45% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BL$	$y$	${y}_{1}$	$BG$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531407187	0.000000012913829	0.250498196636003	0.250498713803289	0.000206455492471
0.349325312001738	0.349325311820970	0.000000051747899	0.250997135452566	0.250998172879910	0.000413322384003
0.412858102914633	0.412858102433137	0.000000116625020	0.251496771788616	0.251498332580776	0.000620601270048
0.487877794992095	0.487877793978929	0.000000207668032	0.251997052439703	0.251999139712905	0.000828292705174
0.576432659704310	0.576432657831064	0.000000324972334	0.252497914287299	0.252500531168605	0.001036397197019
0.680928206687639	0.680928203496700	0.000000468616015	0.252999282395326	0.253002432021838	0.001244915196114
0.804180124924089	0.804180119788193	0.000000638650042	0.253501068081790	0.253504753599681	0.001453847085990
0.949482624793687	0.949482616864799	0.000000835074516	0.254003166434270	0.254007390997589	0.001663193171467
1.120678677500689	1.120678665645472	0.000001057860477	0.254505453598243	0.254510220367458	0.001872953662402
1.322241270688331	1.322241253408206	0.000001306881374	0.255007783602803	0.255013095743029	0.002083128660464
1.559362384105503	1.559362359436883	0.000001581968392	0.255509984618200	0.255515845297051	0.002293718133921
1.838043076134275	1.838043041527667	0.000001882796378	0.256011854994581	0.256018267379588	0.002504721903083
2.165197443921075	2.165197396092670	0.000002208962740	0.256513158255958	0.256520125511436	0.002716139603035
2.548740309797457	2.548740244553857	0.000002559837128	0.257013618311200	0.257021143594555	0.002927970667000
2.997691791894467	2.997691703922826	0.000002934645961	0.257512913057984	0.257520999515239	0.003140214274868
3.522238042881883	3.522237925510326	0.000003332300529	0.258010668878118	0.258019319638710	0.003352869332985
4.133803573914338	4.133803418836006	0.000003751468321	0.258506452996558	0.258515671167109	0.003565934406612
4.845038376786851	4.845038173760608	0.000004190394939	0.258999767706050	0.259009556363199	0.003779407694279
5.669794856541051	5.669794593068782	0.000004646945358	0.259490042275745	0.259500404457730	0.003993286946403

Table III Calculating the effect of a 90% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BL$	$y$	${y}_{1}$	$BG$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531438413	0.000000002347959	0.250498196636003	0.250498290666260	0.000037537298853
0.349325312001738	0.349325311968871	0.000000009408673	0.250997135452566	0.250997324075081	0.000075149270214
0.412858102914633	0.412858102827089	0.000000021204454	0.251496771788616	0.251497055567568	0.000112836021617
0.487877794992095	0.487877794807884	0.000000037757641	0.251997052439703	0.251997431941350	0.000150597653068
0.576432659704310	0.576432659363722	0.000000059085538	0.252497914287299	0.252498390079866	0.000188434256167
0.680928206687639	0.680928206107472	0.000000085202329	0.252999282395326	0.252999855048859	0.000226345912001
0.804180124924089	0.804180123990297	0.000000116117247	0.253501068081790	0.253501738167981	0.000264332689555
0.949482624793687	0.949482623352084	0.000000151830343	0.254003166434270	0.254003934526239	0.000302394643486
1.120678677500689	1.120678675345217	0.000000192336271	0.254505453598243	0.254506320270274	0.000340531811161
1.322241270688331	1.322241267546526	0.000000237612067	0.255007783602803	0.255008749430019	0.000378744210298
1.559362384105503	1.559362379620356	0.000000287626993	0.255509984618200	0.255511050176175	0.000417031834349
1.838043076134275	1.838043069842251	0.000000342321902	0.256011854994581	0.256013020858872	0.000455394649886
2.165197443921075	2.165197435225131	0.000000401623623	0.256513158255958	0.256514425001530	0.000493832589761
2.548740309797457	2.548740297935175	0.000000465417472	0.257013618311200	0.257014986511761	0.000532345550146
2.997691791894467	2.997691775899899	0.000000533562803	0.257512913057984	0.257514383285165	0.000570933381159
3.522238042881883	3.522238021541992	0.000000605861685	0.258010668878118	0.258012241700533	0.000609595883199
4.133803573914338	4.133803545718830	0.000000682071793	0.258506452996558	0.258508128978669	0.000648332794640
4.845038376786851	4.845038339873755	0.000000761874184	0.258999767706050	0.259001547406863	0.000687143787140
5.669794856541051	5.669794808638052	0.000000844880637	0.259490042275745	0.259491926247279	0.000726028451004

Table IV: Calculating the effect of a 99% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BL$	$y$	${y}_{1}$	$BG$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531444658	0.000000000234792	0.250498196636003	0.250498206039026	0.000003753728609
0.349325312001738	0.349325311998452	0.000000000940856	0.250997135452566	0.250997154314805	0.000007514921928
0.412858102914633	0.412858102905879	0.000000002120421	0.251496771788616	0.251496800166482	0.000011283590693
0.487877794992095	0.487877794973674	0.000000003775738	0.251997052439703	0.251997090389816	0.000015059744864
0.576432659704310	0.576432659670251	0.000000005908540	0.252497914287299	0.252497961866475	0.000018843393644
0.680928206687639	0.680928206629623	0.000000008520222	0.252999282395326	0.252999339660562	0.000022634545095
0.804180124924089	0.804180124830710	0.000000011611698	0.253501068081790	0.253501135090250	0.000026433206052
0.949482624793687	0.949482624649527	0.000000015183007	0.254003166434270	0.254003243243258	0.000030239382040
1.120678677500689	1.120678677285142	0.000000019233585	0.254505453598243	0.254505540265181	0.000034053076712
1.322241270688331	1.322241270374152	0.000000023761155	0.255007783602803	0.255007880185195	0.000037874291913
1.559362384105503	1.559362383656989	0.000000028762632	0.255509984618200	0.255510091173597	0.000041703026878
1.838043076134275	1.838043075505074	0.000000034232097	0.256011854994581	0.256011971580533	0.000045539278317
2.165197443921075	2.165197443051483	0.000000040162268	0.256513158255958	0.256513284929952	0.000049383039490
2.548740309797457	2.548740308611232	0.000000046541632	0.257013618311200	0.257013755130601	0.000053234299947
2.997691791894467	2.997691790295015	0.000000053356123	0.257512913057984	0.257513060079946	0.000057093044717
3.522238042881883	3.522238040747901	0.000000060585952	0.258010668878118	0.258010826159496	0.000060959253866
4.133803573914338	4.133803571094798	0.000000068206921	0.258506452996558	0.258506620593791	0.000064832901132
4.845038376786851	4.845038373095557	0.000000076187103	0.258999767706050	0.258999945675031	0.000068713953734
5.669794856541051	5.669794851750771	0.000000084487716	0.259490042275745	0.259490230671667	0.000072602370677

Table V: Calculating the effect of a 99.99% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BL$	$y$	${y}_{1}$	$BG$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531445345	0.000000000002348	0.250498196636003	0.250498196730034	0.000000037537287
0.349325312001738	0.349325312001705	0.000000000009407	0.250997135452566	0.250997135641188	0.000000075149200
0.412858102914633	0.412858102914546	0.000000000021204	0.251496771788616	0.251496772072395	0.000000112835908
0.487877794992095	0.487877794991911	0.000000000037764	0.251997052439703	0.251997052819204	0.000000150597435
0.576432659704310	0.576432659703970	0.000000000059090	0.252497914287299	0.252497914763091	0.000000188433907
0.680928206687639	0.680928206687059	0.000000000085208	0.252999282395326	0.252999282967978	0.000000226345408
0.804180124924089	0.804180124923155	0.000000000116119	0.253501068081790	0.253501068751874	0.000000264331974
0.949482624793687	0.949482624792245	0.000000000151833	0.254003166434270	0.254003167202359	0.000000302393729
1.120678677500689	1.120678677498533	0.000000000192349	0.254505453598243	0.254505454464912	0.000000340530626
1.322241270688331	1.322241270685189	0.000000000237622	0.255007783602803	0.255007784568626	0.000000378742744
1.559362384105503	1.559362384101018	0.000000000287637	0.255509984618200	0.255509985683753	0.000000417030053
1.838043076134275	1.838043076127983	0.000000000342337	0.256011854994581	0.256011856160440	0.000000455392548
2.165197443921075	2.165197443912378	0.000000000401654	0.256513158255958	0.256513159522697	0.000000493830132
2.548740309797457	2.548740309785595	0.000000000465444	0.257013618311200	0.257013619679394	0.000000532342696
2.997691791894467	2.997691791878473	0.000000000533569	0.257512913057984	0.257013619679394	0.000000570930097
3.522238042881883	3.522238042860543	0.000000000605860	0.258010668878118	0.257512914528202	0.000000609592144
4.133803573914338	4.133803573886143	0.000000000682065	0.258506452996558	0.258010670450930	0.000000648328572
4.845038376786851	4.845038376749939	0.000000000761866	0.258999767706050	0.258999769485739	0.000000687139036
5.669794856541051	5.669794856493149	0.000000000844865	0.259490042275745	0.259490044159703	0.000000726023167

Table VI: Calculating the effect of a 101% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BG$	$y$	${y}_{1}$	$BL$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531446046	0.000000000234792	0.250498196636003	0.250498187232982	0.000003753728343
0.349325312001738	0.349325312005025	0.000000000940872	0.250997135452566	0.250997116590330	0.000007514920822
0.412858102914633	0.412858102923388	0.000000002120448	0.251496771788616	0.251496743410756	0.000011283588155
0.487877794992095	0.487877795010516	0.000000003775761	0.251997052439703	0.251997014489601	0.000015059740371
0.576432659704310	0.576432659738369	0.000000005908540	0.252497914287299	0.252497866708141	0.000018843386565
0.680928206687639	0.680928206745655	0.000000008520189	0.252999282395326	0.252999225130115	0.000022634534848
0.804180124924089	0.804180125017468	0.000000011611685	0.253501068081790	0.253501001073366	0.000026433192103
0.949482624793687	0.949482624937846	0.000000015182972	0.254003166434270	0.254003089625328	0.000030239363770
1.120678677500689	1.120678677716235	0.000000019233566	0.254505453598243	0.254505366931365	0.000034053053592
1.322241270688331	2.548740309797457	0.000000023761138	0.255007783602803	0.255007687020483	0.000037874263265
1.559362384105503	1.559362384554016	0.000000028762604	0.255509984618200	0.255509878062891	0.000041702992182
1.838043076134275	1.838043076763475	0.000000034232073	0.256011854994581	0.256011738408736	0.000045539236903
2.165197443921075	2.165197444790666	0.000000040162206	0.256513158255958	0.256513031582089	0.000049382990734
2.548740309797457	2.548740310983681	0.000000046541545	0.257013618311200	0.257013481491945	0.000053234243294
2.997691791894467	2.997691793493916	0.000000053356020	0.257512913057984	0.257512766036189	0.000057092979594
3.522238042881883	3.522238045015861	0.000000060585876	0.258010668878118	0.258010511596931	0.000060959179575
4.133803573914338	4.133803576733874	0.000000068206836	0.258506452996558	0.258506285399542	0.000064832817127
4.845038376786851	4.845038380478142	0.000000076187030	0.258999767706050	0.258999589737314	0.000068713859386
5.669794856541051	5.669794861331324	0.000000084487591	0.259490042275745	0.259489853880096	0.000072602265319

Table VII: Calculating the effect of a 105% decrease in the intraspecific coefficient ${\beta }_{2}$ of the second plant species on the level of biodiversity, applying the ODE 45 numerical approach with equal initial conditions.

$x$	${x}_{1}$	$BG$	$y$	${y}_{1}$	$BL$
0.250000000000000	0.250000000000000	0.000000000000000	0.250000000000000	0.250000000000000	0.000000000000000
0.295533531445352	0.295533531448821	0.000000001173998	0.250498196636003	0.250498149620902	0.000018768638878
0.349325312001738	0.349325312018172	0.000000004704360	0.250997135452566	0.250997041141414	0.000037574592788
0.412858102914633	0.412858102958405	0.000000010602254	0.251496771788616	0.251496629899380	0.000056417915324
0.487877794992095	0.487877795084201	0.000000018878815	0.251997052439703	0.251996862689308	0.000075298656452
0.576432659704310	0.576432659874604	0.000000029542721	0.252497914287299	0.252497676391688	0.000094216861792
0.680928206687639	0.680928206977722	0.000000042601059	0.252999282395326	0.252998996069531	0.000113172571755
0.804180124924089	0.804180125390984	0.000000058058492	0.253501068081790	0.253500733040023	0.000132165820763
0.949482624793687	0.949482625514486	0.000000075914943	0.254003166434270	0.254002782390027	0.000151196635949
1.120678677500689	1.120678678578421	0.000000096167828	0.254505453598243	0.254505020264441	0.000170265035906
1.322241270688331	1.322241272259228	0.000000118805605	0.255007783602803	0.255007300691938	0.000189371029332
1.559362384105503	1.559362386348067	0.000000143812906	0.255509984618200	0.255509451842544	0.000208514612866
1.838043076134275	1.838043079280273	0.000000171160184	0.256011854994581	0.256011272066418	0.000227695769608
2.165197443921075	2.165197448269025	0.000000200810806	0.256513158255958	0.256512524887863	0.000246914465887
2.548740309797457	2.548740315728567	0.000000232707481	0.257013618311200	0.257012934216383	0.000266170649659
2.997691791894467	2.997691799891706	0.000000266779875	0.257512913057984	0.257512177950688	0.000285464245905
3.522238042881883	3.522238053551763	0.000000302928989	0.258010668878118	0.258009882474101	0.000304795154614
4.133803573914338	4.133803588012000	0.000000341033683	0.258506452996558	0.258505615013652	0.000324163244870
4.845038376786851	4.845038395243273	0.000000380934489	0.258999767706050	0.258998877864816	0.000343568352427
5.669794856541051	5.669794880492375	0.000000422437232	0.259490042275745	0.259489100300236	0.000363010272208

Tables I-VII illustrate the impact of varying the intra-specific coefficient ${\beta }_{2}$ of the second plant species y on the extent of biodiversity. Considering Tables I-V, the third and sixth columns calculate the phenomenon change on the second plant species y and the first plant species x respectively at time interval of twenty (20) days. It is observed that when the intra-specific coefficient ${\beta }_{2}=0.000015$ is decreased by 40%, there is an increase on the population biomass of the second plant species y with values presented as ${y}_{1}$. This increase is calculated as BG (Biodiversity Gain) while the population biomass of the first plant species x suffers a loss, BL (Biodiversity Loss). This result is presented on Table I. A further decrease on the value of the intra-specific coefficient ${\beta }_{2}$ by 45%, 90%, 99% and 99.99% give a serious gain on the population of the second plant species y. The percentage biodiversity gain (BG) on the 20th day gives 0.003993286946403, 0.000726028451004, 0.000072602370677 and 0.000000726023167 respectively while the first plant species population suffers a severe loss. The percentage biodiversity loss (BL) at the 20th day gives 0.000004646945358, 0.000000844880637, 0.000000084487716 and 0.000000000844865 respectively. On the other hand, a slight increase of the intra-specific coefficient ${\beta }_{2}$ by 101% and 105% results in a decrease on the population biomass of y by 0.000072602265319 and 0.000363010272208 respectively while the population biomass of the first plant species x rises and is calculated as 0.000000084487591 and 0.000000422437232 respectively at the 20th day.

Comparing this result with that obtained in our previous work (Udoh and Ekaka-a 2021), we have found, in terms of contribution to the knowledge base formed by the process of numerical mathematics on the quantification of a biodiversity scenario, that a specified decrease of the intra-species coefficient of a plant species generally produces a biodiversity gain scenario which shows a positive interaction between the plants of the same species. At some points (101% variation), the biodiversity gain changes to a biodiversity loss which means that there is a competition among the plant species. This can result to the extinction of the plant species. In our next work, we will consider the impact of a specified decrease of the inter-species coefficient of a plant species on a biodiversity gain scenario.

Funding:

Not applicable.

Conflicts of Interests/Competing Interests:

Not applicable.

Availability of Data and Materials

The article contains the data that were used in this study to support its conclusions.

Code Availability:

MATLAB was used for all of the numerical calculations in this work. On request, the codes can be given available.

Author Contribution

The manuscript is single authored. Consequently, the author U.N. prepared it single handedly.

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No competing interests reported.

Numerical Analysis of the Impact of Decreasing the Intra-specific Competition of the Second Plant Specie on Biodiversity Scenario

Status:

Version 1

Abstract

INTRODUCTION

MODEL EQUATIONS

METHOD OF ANALYSIS

RESULTS

DISCUSSION OF RESULTS

CONCLUSION

Declarations