Persistence metrics for a river population in a two-dimensional benthic-drift model

Yu Jin; Qihua Huang; Julia Blackburn; Mark A. Lewis; Yu Jin; Qihua Huang; Julia Blackburn; Mark A. Lewis

doi:10.3934/math.2019.6.1768

AIMS Mathematics

2019, Volume 4, Issue 6: 1768-1795. doi: 10.3934/math.2019.6.1768

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Research article Special Issues

Persistence metrics for a river population in a two-dimensional benthic-drift model

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA
2.
School of Mathematical and Statistical Sciences, Southwest University, Chongqing 400715, China
3.
School of Mathematical and Statistical Sciences, Hubei University of Science and Technology, Xianning 437100, China
4.
Department of Civil and Environmental Engineering, University of Alberta Edmonton, AB, T6G 1H9, Canada
5.
Center for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
6.
Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2E9, Canada

Received: 24 April 2019 Accepted: 19 November 2019 Published: 31 December 2019
MSC : 35K10, 47A75, 92B05

The study of population persistence in river ecosystems is key for understanding population dynamics, invasions, and instream flow needs. In this paper, we extend theories of persistence measures for population models in one-dimensional rivers to a benthic-drift model in two-dimensional depth-averaged rivers. We define the fundamental niche and the source and sink metric, and establish the net reproductive rate R₀ to determine global persistence of a population in a spatially heterogeneous two-dimensional river. We then couple the benthic-drift model into the two-dimensional computational river model, River2D, to study the growth and persistence of a population and its source and sink regions in a river. The theory developed in this study extends existing R₀ analysis to spatially heterogeneous two-dimensional models. The River2D program provides a method to analyze the impact of river morphology on population persistence in a realistic river. The theory and program derived here can be applied to species in real rivers.

Keywords:

Citation: Yu Jin, Qihua Huang, Julia Blackburn, Mark A. Lewis. Persistence metrics for a river population in a two-dimensional benthic-drift model[J]. AIMS Mathematics, 2019, 4(6): 1768-1795. doi: 10.3934/math.2019.6.1768

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Abstract

1. Introduction

Dication magnesium complexes have a unique place in chemistry and biochemistry due to their diverse structural arrangements and applications in numerous fields. Magnesium is very crucial for living beings. For example, about 300 metabolic activities and 800 proteins require this metal to function in our body. This metal is also necessary for DNA [1],[2], RNA [3],[4], antioxidant glutathione production [5], energy generation, oxidative phosphorylation, and glycolysis among other elements, helps the anatomical formation of bone, with exterior layers assisting in the maintenance of blood magnesium levels [6], while magnesium shortage has been associated with decreased bone mass [7],[8]. In plants, Chlorophyll, a magnesium coordination molecule, is required for plant life (Photosynthesis) and the survival of life on Earth [9],[10]. Magnesium also participates in active Ca and K ion transport across cell membranes, which are required for nerve impulse transmission, muscle contraction, and a regular heart rhythm [11],[12].

Computational chemistry has recently acquired popularity among scholars and researchers as a method of addressing real-world challenges in chemical, pharmaceutical, biotechnology, and material science [13]. By Bock et al., the stereochemistry of ligand binding by bivalent magnesium metal was effectively analyzed, as was how likely these ligands are to be water [14]. Ab initio molecular orbital (MO) calculation of M²⁺(H₂O)_n complexes having central alkaline earth metal ion with varying water ligands from one to six at RHF and MP2 level with basis set 6-31+G∗ was investigated by Glendening et al. [15]. The alkaline-earth metal ions M²⁺(H₂O)_n, n = 5-7 (M = Mg, Ca, Sr, and Ba) hydration energies and geometries were determined by Rodriguez-Cruz et al. [16] using the DFT-B3LYP method. According to Pavlov et al. [17], charge transfer between the ligands and the metal lowers the interaction energy of the complex [Mg(H₂O)_n]²⁺ between M···H₂O as the number of ligands rises in the 2^nd coordination sphere. By employing the kinetic energy release measurement method, Bruzzi et al. [18] investigated the binding energies of complexes [Mg(NH₃)_n]²⁺, [Ca(NH₃)_n]²⁺, and [Sr(NH₃)_n]²⁺ for n = 4–20, and these results are supported by DFT calculations. These desired impacts have provided new insights into the trustworthiness of computational methodologies.

Figure 1. Structural representation of the octahedral magnesium complex [Mg(H₂O)₆]²⁺ showing bond lengths.

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Several preceding studies reveal the focus on the coordinative behavior of metal ions however, the natural bond orbital analysis of metal complexes by using the DFT-B3LYP approach with varying numbers of ligands in 2^nd coordination sphere is yet to be studied. The charge transfer from surrounding ligands to a central metal ion and vice versa affects its geometry and other physical features of complexes. Two charge transfer mechanisms, Mulliken and natural, are explained by population analysis. NBO analysis provides a solid framework for researching charge transfer and conjugative interactions in atoms and molecules. The second-order perturbation hypothesis leads to the availability of some electron donor orbitals, acceptor orbitals, and association stabilization energies [19]. The system conjugation increases with increasing E⁽²⁾ value and electron donor interaction. When electron density delocalizes between occupied Lewis type (bond or lone pair) and nominally vacant non-Lewis type (antibonding or Rydberg) NBO orbitals, a stable donor-acceptor interaction is ensured [17]. The natural bond orbital study of magnesium compounds using the DFT-B3LYP approach with varying quantities of ligands in 2^nd coordination sphere remains unexplored. This work reports the study of natural transfer properties in the coordination complex [Mg(H₂O)₆]²⁺. Figure 1 illustrates the optimized structural representation of complex [Mg(H₂O)₆]²⁺ that displays bond lengths. The central metal ion is attached to identical six water ligands, forming an octahedral structure. The distance between Mg and O is 2.112 Å, while the distance between O and H is 0.967Å. The remaining four complexes were created by gradually expanding the number of water molecules within 2^nd coordination sphere from one to four. Natural charge transfer behavior of the different complexes was then compared. This study of magnesium complexes will shed light on the charge transfer behavior of metal ions. The analysis of these complexes will be beneficial for medicinal design and could aid in the development of new drugs.

2. Materials and methods

The quantum computations in this extensive analysis of metal complexes were accomplished with the Gaussian 16 program packages [20]. Gaussian inputs of magnesium ion compounds were generated, and outputs were displayed using GaussView 6 [21]. The structural geometrics of complexes were firstly optimized utilizing various basis sets with B3LYP functional, which combines Becke's gradient-correlated exchange functional (B3) [22] and Lee-Yang-Parr (LYP) [23]. These inputs were acquired by retaining the ligands in a quasi-octahedral structure. There was no symmetry limitation enforced, and the C₁ point group symmetry was used for optimization. Frequency estimations were carried out during the geometry optimization process, and global minima were validated. Every calculation was carried out by applying 6-311++G(d,p) basis set and B3LYP functional. NBOs provide precise details on the type of electronic conjugation occurring between molecular bonds. In metal complexes, the delocalization of electrons results once the hybridized orbitals of water molecules and metal ions coincides. NBO evaluation is a strong approach for determining this electron delocalization. NBOs strongly support the assumption that localized bonds and lone pairs are the essential building blocks of molecular structure, hence it is feasible to understand ab initio wave functions in perspective of Lewis structure theories by effectively converting them to NBO form. The NBO technique was utilized to study how the non-bonding pairs of oxygen atoms in the water decreased their native charge densities. NBO analysis is performed by looking at all conceivable interactions between ‘full’ (donor) Lewis-type NBOs and ‘empty’ (acceptor) non-Lewis NBOs and evaluating their energetic significance using second-order perturbation theory. Since these exchanges result in the transfer of occupancy from the idealized Lewis structure's localized NBOs into the unoccupied non-Lewis orbitals (and hence deviations from the idealized Lewis structure description), they are termed to as “delocalization” corrections to the zeroth-order natural Lewis structure [23]. This work presents the outcomes of a second-order perturbation theory investigation of the Fock matrix within the NBO of the complexes. The delocalization-related stabilizing energy E⁽²⁾ for each donating NBO (i) and receiver NBO (j) is determined as

$E^{(2)} = q_{i} \frac{< i | \hat{F} | j >}{ε_{i} - ε_{j}}$

where q_i stands for orbital occupancy of a donor, ε_i, ε_j represents orbital energies, and F(i,j) for the off-diagonal NBO Fock matrix component [24]. Higher value of stabilization energy E⁽²⁾ denotes powerful interaction between acceptors and donors [25].

3. Results and discussion

Figure 2. Natural charges on constituent atoms in the complex [Mg(H₂O)₆]²⁺.

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Table 1. Natural charges on the central metal ion Mg^Q and ligands to metal charge transfer ΔQ in the complexes [[Mg(H₂O)₆](H₂O)_n]²⁺; n = 0-4.

Complex	Mg^Q(e)	ΔQ(e)
[Mg(H₂O)₆]²⁺	1.821	0.179
[[Mg(H₂O)₆](H₂O)]²⁺	1.820	0.180
[[Mg(H₂O)₆](H₂O)₂]²⁺	1.819	0.181
[[Mg(H₂O)₆](H₂O)₃]²⁺	1.818	0.182
[[Mg(H₂O)₆](H₂O)₄]²⁺	1.817	0.183

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The natural charges on the central metal ion Mg²⁺ and charge transfer in the complexes [[Mg(H₂O)₆](H₂O)_n]²⁺; n = 0-4 are demonstrated in Table 1. The charge transfer takes place in all complexes. The NBO partial charge on metal ions does not widely vary when the number of molecules within the 2^nd coordination sphere is increased, but there are modest increases, as shown in Table 1. Figure 2 depicts the octahedral structural representation of complex [Mg(H₂O)₆]²⁺ with natural charges on constituent atoms. This central metal ion is surrounded by six identical water molecules in the 1^st coordination sphere. The total charge constituent in the whole complex equals +2e. Figure 3(a) shows the representation of the complex when one water ligand is added to its 2^nd coordination sphere. Likewise, Figure 3(b), Figure 3(c), and Figure 3(d) represent the figurative representatives of complexes after the addition of two, three, and four water ligands to their 2^nd coordination sphere, respectively. The total electron density of the [Mg(H₂O)₆]²⁺ is used to express the overall effectiveness of the natural Lewis structure analysis as a percentage. Table 2 demonstrates the importance of valance non-Lewis orbitals in comparison to extra-valence electron shells in modest deviations from a confined Lewis structure model.

Figure 3. The natural charges on constituent atoms in the complex (a) [[Mg(H₂O)₆](H₂O)]²⁺, (b) [[Mg(H₂O)₆](H₂O)₂]²⁺, (c) [[Mg(H₂O)₆](H₂O)₃]²⁺, and (d) [[Mg(H₂O)₆](H₂O)₄]²⁺ with extension of one to four ligands in the 2^nd coordination sphere respectively.

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Table 2. Separation of Lewis and non-Lewis occupancies within complex [Mg(H₂O)₆]²⁺ into core, valence, and Rydberg shell components.

Orbitals	Occupancy
Core	21.99748 (99.989% of 22)
Valence Lewis	47.77540 (99.532% of 48)
Total Lewis	69.77288 (99.676% of 70)
Valence non-Lewis	0.18649 (0.266% of 70)
Rydberg non-Lewis	0.04063 (0.058% of 70)
Total non-Lewis	0.22712 (0.324% of 70)

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The electron delocalization through lone pairs of oxygen to iron orbitals was evaluated. The table below shows the two most powerful interactions. The interchange of non-bonding pairs of oxygen with metal n∗ orbitals is determined to be greatest in the instance of a complex [Mg(H₂O)₆]²⁺. The greatest interaction in the complex is likewise demonstrated to be between metal orbitals and virtual orbitals.

Table 3. Result of second-order perturbation theory evaluation of Fock matrix within NBO of complex [Mg(H₂O)₆]²⁺.

Donor NBOs	Occupancy (e)	Hybrid (%)	Acceptor NBOs	Occupancy (e)	Hybrid (%)	E⁽²⁾ (kcal/mol)
LP O (2)	1.96849	s47.65p52.34	LP∗ Mg	0.17178	s100	22.67
LP O (5)	1.96853	s47.66p52.34	LP∗ Mg	0.17178	s100	22.63
LP O (8)	1.96853	s47.65p52.34	LP∗ Mg	0.17178	s100	22.63
LP O (11)	1.96851	s47.66p52.34	LP∗ Mg	0.17178	s100	22.66
LP O (14)	1.96849	s47.65p52.34	LP∗ Mg	0.17178	s100	22.67
LP O (17)	1.96854	s47.66p52.34	LP∗ Mg	0.17178	s100	22.62

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From Table 3, it is seen that delocalization of oxygen lone pairs to n∗ orbitals of Mg²⁺ occurred in the stronger interactions of LP O (2) and LP O (14) with LP∗ Mg with occupancy of 0.17178e stabilizes [Mg(H₂O)₆]²⁺ complex by 22.67 kcal/mol. These stabilization energies of complexes are balanced by the remaining stabilization energies of complexes. This analysis reveals that, on the donor orbital side, the p orbital contributes more than the s orbital, whereas the d orbital does not contribute. On the acceptor side, the contribution of the p and d orbitals was negligible in comparison to the s orbital for the maximum stabilization energy. In a few cases of acceptor orbitals, the contribution of the p orbital and d orbital is also seen. The greatest stabilization energy of complex [Mg(H₂O)₆]²⁺ is less than that of the complex [Zn(H₂O)₆]²⁺ as reported by Pokharel et al. [26].

The NBO partial charge on the metal ions does not change much on adding the number of ligands within the 2^nd coordination sphere. It displays some findings from an investigation into the Fock matrix within the NBO of complexes by utilizing second-order perturbation analysis. Only the interaction resulting from the delocalization of the electrons from the oxygen lone pairs of ligands in the 1^st coordination sphere to the n∗ orbitals of Mg²⁺ was obtained to have stabilization energies of more than 5 kcal/mol in all circumstances. As a result, these tables only include the most powerful interactions. The powerful interaction among the associated interactions is one in which electrons from non-binding pairs of O (2) are delocalized to the LP∗ Mg with occupancy 0.17343e, as shown in Table 4. This is the interaction when one water ligand is attached within the 2^nd coordination sphere of a complex [Mg(H₂O)₆]²⁺. As shown by the preceding result, the bond between magnesium and oxygen has been shortened compared to that of the complex [Mg(H₂O)₆]²⁺ because of the presence of one water ligand in the 2^nd coordination sphere.

Table 4. Result of second-order perturbation theory evaluation of matrix within NBO of the complex [[Mg(H₂O)₆](H₂O)]²⁺.

Donor NBOs	Occupancy (e)	Hybrid (%)	Acceptor NBOs	Occupancy (e)	Hybrid (%)	E⁽²⁾ (kcal/mol)
LP O (2)	1.96238	s43.80p56.20	LP∗ Mg	0.17343	s100	25.47
LP O (5)	1.96938	s47.66p52.34	LP∗ Mg	0.17343	s100	22.38
LP O (8)	1.96955	s47.65p52.35	LP∗ Mg	0.17343	s100	22.06
LP O (11)	1.96874	s46.67p53.32	LP∗ Mg	0.17343	s100	22.51
LP O (14)	1.96938	s47.66p52.34	LP∗ Mg	0.17343	s100	22.37
LP O (17)	1.96967	s47.74p52.25	LP∗ Mg	0.17343	s100	22.15

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Table 5. Result of second-order perturbation theory evaluation of Fock matrix within NBO of the complex [[Mg(H₂O)₆](H₂O)₂]²⁺.

Donor NBOs	Occupancy (e)	Hybrid (%)	Acceptor NBOs	Occupancy (e)	Hybrid (%)	E⁽²⁾ (kcal/mol)
LP O (2)	1.96371	s44.71p55.79	LP∗ Mg	0.17413	s99.99p0.01	24.64
LP O (5)	1.97016	s46.32p53.67	LP∗ Mg	0.17413	s99.99p0.01	24.71
LP O (8)	1.96726	s44.71p55.28	LP∗ Mg	0.17413	s99.99p0.01	22.82
LP O (11)	1.97003	s46.70p53.29	LP∗ Mg	0.17413	s99.99p0.01	21.86
LP O (14)	1.97015	s46.34p53.66	LP∗ Mg	0.17413	s99.99p0.01	21.74
LP O (17)	1.96663	s45.22p54.77	LP∗ Mg	0.17413	s99.99p0.01	23.47

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Table 5 shows two strong E⁽²⁾ that are near to one another. One is from the delocalization of electrons from non-bonding pairs of O (5) and another is from O (2) to the LP∗ Mg having occupancy 0.17413e with stabilization energies of 24.71 kcal/mol and 24.64 kcal/mol respectively.

In a complex [[Mg(H₂O)₆](H₂O)₃]²⁺, the most powerful engagement is because of the delocalization of the electrons from non-bonding pairs of O (2) to LP∗ Mg with occupancy 0.17514e that balanced this ion by 24.09 kcal/mol. Moreover, the other two nearly equal interactions are seen because of the delocalization of non-bonding pairs of O (8) and O (17) with LP∗ Mg having stabilization energy of 23.06 kcal/mol and 23.13 kcal/mol respectively. The powerful interactions are presented in Table 6.

Table 6. Result of second-order perturbation theory evaluation of Fock matrix within NBO of the complex [[Mg(H₂O)₆](H₂O)₃]²⁺.

Donor NBOs	Occupancy (e)	Hybrid (%)	Acceptor NBOs	Occupancy (e)	Hybrid (%)	E⁽²⁾ (kcal/mol)
LP O (2)	1.96511	s41.70p58.30d0.00	LP∗ Mg	0.17514	s99.99p0.01	24.09
LP O (5)	1.96762	s43.30p56.70d0.01	LP∗ Mg	0.17514	s99.99p0.01	22.55
LP O (8)	1.96754	s44.71p55.29d0.01	LP∗ Mg	0.17514	s99.99p0.01	23.06
LPO (11)	1.97130	s46.77p53.23d0.01	LP∗ Mg	0.17514	s99.99p0.01	21.21
LPO (14)	1.96764	s43.23p56.76d0.01	LP∗ Mg	0.17514	s99.99p0.01	22.57
LP O (17)	1.96763	s45.06p54.94d0.00	LP∗ Mg	0.17514	s99.99p0.01	23.13

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Lastly, in Table 7, the greatest interactions among strong interactions are because of the delocalization of the electrons from the non-bonding pairs of O (8) with LP∗ Mg having occupancy of 0.17653e by stabilization energies of 25.21 kcal/mol. The stabilization energies of the remaining three interactions, which are strong, are 23.13 kcal/mol, 22.92 kcal/mol, and 22.16 kcal/mol.

Table 7. Result of second-order perturbation theory evaluation of Fock matrix within NBO of the complex [[Mg(H₂O)₆](H₂O)₄]²⁺.

Donor NBOs	Occupancy (e)	Hybrid (%)	Acceptor NBOs	Occupancy (e)	Hybrid (%)	E⁽²⁾ (kcal/mol)
LP O (2)	1.96618	s41.20p58.80d0.00	LP∗ Mg	0.17653	s99.99p0.00d0.01	23.17
LP O (5)	1.96830	s43.07p56.93d0.02	LP∗ Mg	0.17653	s99.99p0.00d0.01	22.16
LP O (8)	1.96308	s42.10p57.90d0.00	LP∗ Mg	0.17653	s99.99p0.00d0.01	25.21
LP O (11)	1.97119	s47.02p52.97d0.01	LP∗ Mg	0.17653	s99.99p0.00d0.01	21.36
LP O (14)	1.96838	s43.03p56.97d0.01	LP∗ Mg	0.17653	s99.99p0.00d0.01	22.06
LP O (17)	1.96827	s44.77p55.23d0.00	LP∗ Mg	0.17653	s99.99p0.00d0.01	22.92

| Show Table

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4. Conclusions

Natural charge transfer takes place between magnesium ion and the water ligands in complexes [Mg(H₂O)₆]²⁺ and [[Mg(H₂O)₆](H₂O)_n]²⁺; n=1-4 has been successfully studied by using NBO analysis. Among these five complexes, ligands to ion charge transfer were discovered to be greatest in complex [[Mg(H₂O)₆](H₂O)₄]²⁺ and smallest in complex [Mg(H₂O)₆]²⁺. The greatest stabilization energy associated with the delocalization of electrons from non-bonding pair of oxygen having LP∗ Mg was found to be 25.47 kcal/mol in complex [[Mg(H₂O)₆](H₂O)₂]²⁺. The number of stronger interactions was observed to increase with the introduction of ligands within 2^nd coordination sphere. The delocalization of electrons from non-bonding pair of oxygen in the initial coordination sphere was greater on which ligand was attached within 2^nd coordination sphere than in the other oxygen lone pair. By adding more ligands to the complex's initial coordination sphere, it is possible to intensify the structures, and these intensified structures will lead to drug design.

References

[1]	J. D. Allan, Stream Ecology: Structure and Function of Running Waters, Chapman & Hall, London, 1995.
[2]	K. E. Anderson, L. R. Harrisonb, R. M. Nisbet, et al. Modeling the influence of flow on invertebrate drift across spatial scales using a 2D hydraulic model and a 1D population model, Ecol. Model., 265 (2013), 207-220. doi: 10.1016/j.ecolmodel.2013.06.011
[3]	K. E. Anderson, A. J. Paul, E. McCauley, et al. Instream flow needs in streams and rivers: The importance of understanding ecological dynamics, Front. Ecol. Environ., 4 (2006), 309-318. doi: 10.1890/1540-9295(2006)4[309:IFNISA]2.0.CO;2
[4]	J. D. Armstrong, P. S. Kemp, G. J. A. Kennedy, et al. Habitat requirements of Atlantic salmon and brown trout in rivers and streams, Fish. Res., 62 (2003), 143-170. doi: 10.1016/S0165-7836(02)00160-1
[5]	K. E. Bencala, R. A. Walters, Simulation of solute transport in a mountain poop-and-riffle stream: A transient storage model, Water Resour. Res., 19 (1983), 718-724. doi: 10.1029/WR019i003p00718
[6]	D. J. Booker, Hydraulic modelling of fish habitat in urban rivers during high flows, Hydrol. Process., 17 (2003), 577-599. doi: 10.1002/hyp.1138
[7]	A. N. Brooks, T. J. R. Hughes, Steamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the imcompressible Navier-Stokes equations, Comput. Method. Appl. Mech. Eng., 32 (1982), 199-259. doi: 10.1016/0045-7825(82)90071-8
[8]	N. J. Clifford, O. P. Harmar, G. Harvey, et al. Physical habitat, eco-hydraulics and river design: A review and re-evaluation of some popular concepts and methods, Aquat. Conserv., 16 (2006), 389-408. doi: 10.1002/aqc.736
[9]	J. M. Cushing, Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Resour. Model., 8 (1994), 297-333. doi: 10.1111/j.1939-7445.1994.tb00188.x
[10]	D. L. DeAngelis, M. Loreaub, D. Neergaardc, et al. Modelling nutrient-periphyton dynamics in streams: The importance of transient storage zones, Ecol. Model., 80 (1995), 149-160. doi: 10.1016/0304-3800(94)00066-Q
[11]	MIKE 21 Flow Model FM: Particle Tracking Module, User Guide, Horsholm, Denmark, 2011, pp. 56.
[12]	MIKE 21 Flow Model FM: Particle Tracking Module, Step-by-step training guide, Horsholm, Denmark, 2011, pp. 48.
[13]	J. M. Elliott, Time spent in the drift by downstream-dispersing invertebrates in a Lake District stream, Freshwater Biol., 47 (2002), 97-106. doi: 10.1046/j.1365-2427.2002.00784.x
[14]	L. Gallien, T. Münkemüller, C. H. Albert, et al. Predicting potential distributions of invasive species: Where to go from here?, Divers. Distrib., 16 (2010), 331-342. doi: 10.1111/j.1472-4642.2010.00652.x
[15]	A. Ghanem, P. M. Steffler, F. E. Hicks, et al. 1995, Dry area treatment for two-dimensional finite element shallow flow modeling, Proceeding of the 12th Canadian Hydrotechnical Conference, Ottawa, Ontario, June, 1995, pp.10.
[16]	A. Ghanem, P. M. Steffler, F. E. Hicks, et al. Two dimensional finite element model for aquatic habitats, Water Resources Engineering Report 95-S1, Department of Civil Engineering, University of Alberta, 1995, pp. 189.
[17]	R. Guenther, J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover, New York, 1996.
[18]	J. W. Hayes, N. F. Hughes, L. H. Kelly, Process-based modelling of invertebrate drift transport, net energy intake and reach carrying capacity for drift-feeding salmonids, Ecol. Model., 207 (2007), 171-188. doi: 10.1016/j.ecolmodel.2007.04.032
[19]	F. M. Hilker, M. A. Lewis, Predator-prey systems in streams and rivers, Theor. Ecol., 3 (2010), 175-193. doi: 10.1007/s12080-009-0062-4
[20]	Q. Huang, Y. Jin, M. A. Lewis, R₀ Analysis of a Benthic-Drift Model for a Stream Population, SIAM J. Appl. Dyn. Syst., 15 (2016), 287-321. doi: 10.1137/15M1014486
[21]	I. Ibanez, E. Gornish, L. Buckley, et al. Moving forward in global-change ecology: Capitalizing on natural variability, Ecol. Evol., 3 (2013), 170-181. doi: 10.1002/ece3.433
[22]	Y. Jin, F. M. Hilker, P. M. Steffler, et al. Seasonal invasion dynamics in a spatially heterogeneous river with fluctuating flows, Bull. Math. Biol., 76 (2014), 1522-1565. doi: 10.1007/s11538-014-9957-3
[23]	Y. Jin, M. A. Lewis, Seasonal influences on population spread and persistence in streams: Critical domain size, SIAM J. Appl. Math., 71 (2011), 1241-1262. doi: 10.1137/100788033
[24]	Y. Jin, M. A. Lewis, Seasonal influences on population spread and persistence in streams: Spreading speeds, J. Math. Biol., 65 (2012), 403-439. doi: 10.1007/s00285-011-0465-x
[25]	M. Krkosěk, M. A. Lewis, An R₀ theory for source-sink dynamics with application to Dreissena competition, Theor. Ecol., 3 (2010), 25-43. doi: 10.1007/s12080-009-0051-7
[26]	J. Lancaster, B. J. Downes, Linking the hydraulic world of individual organisms to ecological processes: Putting ecology into ecohydraulics, River Res. Appl., 26 (2010), 385-403. doi: 10.1002/rra.1274
[27]	L. B. Leopold, W. B. Langbein, River Meanders, Sci. Am., 214 (1966), 60-73.
[28]	F. Lutscher, M. A. Lewis, E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160. doi: 10.1007/s11538-006-9100-1
[29]	F. Lutscher, R. M. Nisbet, E. Pachepsky, Population persistence in the face of advection, Theor. Ecol., 3 (2010), 271-284. doi: 10.1007/s12080-009-0068-y
[30]	D. A. Lytle, D. M. Merritt, Hydrologic regimes and riparian forests: A structured population model for cottonwood, Ecology, 85 (2004), 2493-2503. doi: 10.1890/04-0282
[31]	H. M. McKenzie, Y. Jin, J. Jacobsen, et al. R₀ Analysis of a Spationtemporal Model for a Stream Population, SIAM J. Appl. Dyn. Syst., 11 (2012), 567-596. doi: 10.1137/100802189
[32]	R. T. Milhous, T. J. Waddle, Physical Habitat Simulation (PHABSIM) Software for Windows (v.1.5.1), Fort Collins, CO: USGS Fort Collins Science Center, 2012.
[33]	A. M. Mouton, M. Schneider, J. Depestele, et al. Fish habitat modelling as a tool for river management, Ecol. Eng., 29 (2007), 305-315. doi: 10.1016/j.ecoleng.2006.11.002
[34]	K. Müller, Investigations on the organic drift in North Swedish streams, Report of the Institute of Freshwater Research, Drottningholm, 34 (1954), 133-148.
[35]	K. Müller, The colonization cycle of freshwater insects, Oecologica, 53 (1982), 202-207.
[36]	T. Nagaya, Y. Shiraishi, K. Onitsuka, et al. Evaluation of suitable hydraulic conditions for spawning of ayu with horizontal 2D numerical simulation and PHABSIM, Ecol. Model., 215 (2008), 133-143. doi: 10.1016/j.ecolmodel.2008.02.043
[37]	S. A. Nazirov, A. A. Abduazizov, Approximate calculation of the multiple integrals' value by repeated application of Gauss and Simpson's quadrature formulas, Appl. Math. Sci., 7 (2013), 4223-4235.
[38]	E. Pachepsky, F. Lutscher, R. M. Nisbet, et al. Persistence, spread and the drift paradox, Theor. Popul. Biol., 67 (2005), 61-73. doi: 10.1016/j.tpb.2004.09.001
[39]	V. B. Pasour, S. P. Ellner, Computational and analytic perspectives on the drift paradox, SIAM J. Appl. Dyn. Syst., 9 (2010), 333-356. doi: 10.1137/09075500X
[40]	N. L. Poff, J. K. H. Zimmerman, Ecological responses to altered flow regimes: A literature review to inform the science and management of environmental flows, Freshwater. Biol., 55 (2010), 194-205. doi: 10.1111/j.1365-2427.2009.02272.x
[41]	D. Speirs, W. Gurney, Population persistence in rivers and estuaries, Ecology, 82 (2001), 1219-1237. doi: 10.1890/0012-9658(2001)082[1219:PPIRAE]2.0.CO;2
[42]	I. Stakgold, Green's Functions and Boundary Value Problems, 2 Eds., Wiley, New York, 1998.
[43]	P. Steffler, J. Blackburn, Two-dimensional depth averaged model of river hydrodynamics and Fish habitat, River2D User's Manual, University of Albert, Canada, 2002.
[44]	T. J. Stohlgren, P. Ma, S. Kumar, et al. Ensemble habitat mapping of invasive plant species, Risk Anal., 30 (2010), 224-235. doi: 10.1111/j.1539-6924.2009.01343.x
[45]	A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall Inc., 1971.
[46]	C. B. Talbert, M.K. Talbert, User Manual for SAHM package for Vis Trails, US Geological Survey, 2012.
[47]	H. Thieme, Spectral bound and reproductive number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2010), 188-211.
[48]	F. Wang, B. Lin, Modelling habitat suitability for fish in the fluvial and lacustrine regions of a new Eco-City, SIAM J. Appl. Math., 267 (2013), 115-126.
[49]	W. Wang, X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942
[50]	L. L. Wehmeyer, C. R. Wagner, Relation between Flows and Dissolved Oxygen in the Roanoke River between Roanoke Rapids Dam and Jamesville, North Carolina, 2005-2009, Scientific Scientific Investigations Report, Department of the Interior, U.S. Geological Survey, 2011. Available from: https://pubs.usgs.gov/sir/2011/5040/pdf/sir2011-5040.pdf.

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