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Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems
A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects
1. | Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85287, USA |
2. | Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA |
We propose and study a two patch Rosenzweig-MacArthur prey-predator model with immobile prey and predator using two dispersal strategies. The first dispersal strategy is driven by the prey-predator interaction strength, and the second dispersal is prompted by the local population density of predators which is referred as the passive dispersal. The dispersal strategies using by predator are measured by the proportion of the predator population using the passive dispersal strategy which is a parameter ranging from 0 to 1. We focus on how the dispersal strategies and the related dispersal strengths affect population dynamics of prey and predator, hence generate different spatial dynamical patterns in heterogeneous environment. We provide local and global dynamics of the proposed model. Based on our analytical and numerical analysis, interesting findings could be summarized as follow: (1) If there is no prey in one patch, then the large value of dispersal strength and the large predator population using the passive dispersal in the other patch could drive predator extinct at least locally. However, the intermediate predator population using the passive dispersal could lead to multiple interior equilibria and potentially stabilize the dynamics; (2) The large dispersal strength in one patch may stabilize the boundary equilibrium and lead to the extinction of predator in two patches locally when predators use two dispersal strategies; (3) For symmetric patches (i.e., all the life history parameters are the same except the dispersal strengths), the large predator population using the passive dispersal can generate multiple interior attractors; (4) The dispersal strategies can stabilize the system, or destabilize the system through generating multiple interior equilibria that lead to multiple attractors; and (5) The large predator population using the passive dispersal could lead to no interior equilibrium but both prey and predator can coexist through fluctuating dynamics for almost all initial conditions.
References:
[1] |
P. Auger and J. Poggiale,
Emergence of population growth models: Fast migration and slow growth, Journal of Theoretical Biology, 182 (1996), 99-108.
doi: 10.1006/jtbi.1996.0145. |
[2] |
D. E. Bowler and T. G. Benton,
Causes and consequences of animal dispersal strategies: Relating individual behaviour to spatial dynamics, Biological Reviews, 80 (2005), 205-225.
doi: 10.1017/S1464793104006645. |
[3] |
J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols,
Dispersal Oxford University Press, New York, (2001). |
[4] |
R. Cressman and K. Vlastimil,
Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, Journal of Mathematical Biology, 67 (2013), 329-358.
doi: 10.1007/s00285-012-0548-3. |
[5] |
G. K. Davoren, W. A. Montevecchi and J. T. Anderson,
Distributional patterns of a marine bird and its prey: Habitat selection based on prey and conspecific behaviour, Marine Ecology Progress Series, 256 (2003), 229-242.
doi: 10.3354/meps256229. |
[6] |
D. F. Fraser and R. D. Cerri,
Experimental evaluation of predator-prey relationships in a patchy environment: consequences for habitat use patterns in minnows, Ecology, 62 (1982), 307-313.
doi: 10.2307/1938947. |
[7] |
J. F. Fujita, M. S. and R. V. Bowen, Ant foraging behavior: Ambient temperature influences
prey selection, Behavioral Ecology and Sociobiology, 15 65–68. |
[8] |
S. Ghosh and S. Bhattacharyya,
A two-patch prey-predator model with food-gathering activity, Journal of Applied Mathematics and Computing, 37 (2011), 497-521.
doi: 10.1007/s12190-010-0446-z. |
[9] |
D. Grünbaum and V. R. Richard,
Black-browed albatrosses foraging on antarctic krill: density-dependence through local enhancement?, Ecology, 84 (2003), 3265-3275.
|
[10] |
I. Hanski,
Metapopulation dynamics, Nature, 396 (1998), 41-49.
|
[11] |
I. Hanski,
Habitat connectivity, habitat continuity, and metapopulations in dynamic landscapes, Oikos, 87 (1999), 209-219.
doi: 10.2307/3546736. |
[12] |
I. Hanski,
Metapopulation Ecology Oxford University Press, Oxford, 1999. |
[13] |
I. Hanski and M. Gilpin,
Metapopulation dynamics: Brief history and conceptual domain, Biological Journal of the Linnean Society, 42 (1991), 3-16.
doi: 10.1016/B978-0-12-284120-0.50004-8. |
[14] |
L. Hansson,
Dispersal and connectivity in metapopulations, Biological Journal of the Linnean Society, 42 (1991), 89-103.
doi: 10.1016/B978-0-12-284120-0.50009-7. |
[15] |
A. Hastings,
Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[16] |
S. Hsu, S. Hubbell and P. Wlatman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[17] |
S. Hsu,
On global stability of a predator-prey system, Mathematical Biosciences, 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[18] |
Y. Huang and O. Diekmann,
Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001), 561-581.
doi: 10.1007/s002850100107. |
[19] |
V. Hutson,
A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.
doi: 10.1007/BF01540776. |
[20] |
R. A. Ims and D. Hjermann, Condition-dependent dispersal, in Dispersal, Oxford University
Press, Oxford, (2001), 203–216. |
[21] |
I. M. J{á}nosi and I. Scheuring,
On the evolution of density dependent dispersal in a spatially structured population model, Journal of Theoretical Biology, 187 (1997), 397-408.
|
[22] |
V. A. Jansen,
Regulation of predator-prey systems through spatial interactions: A possible solution to the paradox of enrichment, Oikos, 74 (1995), 384-390.
doi: 10.2307/3545983. |
[23] |
V. A. Jansen,
The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001), 119-131.
doi: 10.1006/tpbi.2000.1506. |
[24] |
Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with dispersal in predators driven by the strength of predation, preprint, arXiv: 1505.03820. |
[25] |
P. Kareiva and G. Odell,
Swarms of predators exhibit ''prey-taxis" if individual predators use area-restricted search, American Naturalist,, 130 (1987), 233-270.
|
[26] |
A. Kiester and M. Slatkin,
A strategy of movement and resource utilization, Theoretical Population Biology, 6 (1974), 1-20.
doi: 10.1016/0040-5809(74)90028-8. |
[27] |
V. Krivan,
Dispersal dynamics: Distribution of lady beetles (coleoptera: Coccinellidae), European Journal of Entomology, 105 (2008), 405-409.
doi: 10.14411/eje.2008.051. |
[28] |
M. Kummel, D. Brown and A. Bruder,
How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.
doi: 10.1111/j.1600-0706.2012.20805.x. |
[29] |
X. Liu and L. Chen,
Complex dynamics of Holling type Ⅱ Lotka--Volterra predator--prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16 (2003), 311-320.
doi: 10.1016/S0960-0779(02)00408-3. |
[30] |
Y. Liu,
The Dynamical Behavior of a two Patch Predator-prey Model Honor Thesis, from The College of William and Mary, 2010. |
[31] |
G. P. Markin,
Foraging behavior of the argentine ant in a california citrus grove, Journal of Economic Entomology, 63 (1970), 740-744.
doi: 10.1093/jee/63.3.740. |
[32] |
M. Massot, J. Clobert, P. Lorenzon and J.-M. Rossi,
Condition-dependent dispersal and ontogeny of the dispersal behaviour: An experimental approach, Journal of Animal Ecology, 71 (2002), 253-261.
doi: 10.1046/j.1365-2656.2002.00592.x. |
[33] |
E. Matthysen,
Density-dependent dispersal in birds and mammals, Ecography, 28 (2005), 403-416.
doi: 10.1111/j.0906-7590.2005.04073.x. |
[34] |
T. Namba,
Density-dependent dispersal and spatial distribution of a population, Journal of Theoretical Biology, 86 (1980), 351-363.
doi: 10.1016/0022-5193(80)90011-9. |
[35] |
D. Nguyen-Ngoc, T. Nguyen-Huu and P. Auger,
Effects of fast density dependent dispersal on pre-emptive competition dynamics, Ecological Complexity, 10 (2012), 26-33.
doi: 10.1016/j.ecocom.2011.12.003. |
[36] |
J. Poggiale,
From behavioural to population level: Growth and competition, Mathematical and Computer Modelling, 27 (1998), 41-49.
doi: 10.1016/S0895-7177(98)00004-1. |
[37] |
J. D. Reeve,
Environmental variability, migration, and persistence in host-parasitoid systems, American Naturalist, 132 (1988), 810-836.
doi: 10.1086/284891. |
[38] |
O. Ronce, I. Olivieri, J. Clobert and E. G. Danchin, Perspectives on the study of dispersal evolution, Oxford University Press, Oxford, 2001.
![]() |
[39] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[40] |
J. F. Savino and R. A. Stein,
Behavioural interactions between fish predators and their prey: effects of plant density, Animal Behaviour, 37 (1989), 311-321.
doi: 10.1016/0003-3472(89)90120-6. |
[41] |
J. A. Silva, M. L. De Castro and D. A. Justo,
Stability in a metapopulation model with density-dependent dispersal, Bulletin of Mathematical Biology, 63 (2001), 485-505.
doi: 10.1006/bulm.2000.0221. |
[42] |
A. T. Smith and M. M. Peacock,
Conspecific attraction and the determination of metapopulation colonization rates, Conservation Biology, 4 (1990), 320-323.
doi: 10.1111/j.1523-1739.1990.tb00294.x. |
[43] |
J. Stamps,
Conspecific attraction and aggregation in territorial species, American Naturalist, 131 (1988), 329-347.
doi: 10.1086/284793. |
[44] |
Foraging behavior of ants: experiments with two species of myrmecine ants, Behavioral Ecology and Sociobiology, 2 (1977), 147–167. |
[45] |
H. R. Thieme,
Mathematics in Population Biology Princeton University Press, 2003. |
show all references
References:
[1] |
P. Auger and J. Poggiale,
Emergence of population growth models: Fast migration and slow growth, Journal of Theoretical Biology, 182 (1996), 99-108.
doi: 10.1006/jtbi.1996.0145. |
[2] |
D. E. Bowler and T. G. Benton,
Causes and consequences of animal dispersal strategies: Relating individual behaviour to spatial dynamics, Biological Reviews, 80 (2005), 205-225.
doi: 10.1017/S1464793104006645. |
[3] |
J. Colbert, E. Danchin, A. A. Dhondt and J. D. Nichols,
Dispersal Oxford University Press, New York, (2001). |
[4] |
R. Cressman and K. Vlastimil,
Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, Journal of Mathematical Biology, 67 (2013), 329-358.
doi: 10.1007/s00285-012-0548-3. |
[5] |
G. K. Davoren, W. A. Montevecchi and J. T. Anderson,
Distributional patterns of a marine bird and its prey: Habitat selection based on prey and conspecific behaviour, Marine Ecology Progress Series, 256 (2003), 229-242.
doi: 10.3354/meps256229. |
[6] |
D. F. Fraser and R. D. Cerri,
Experimental evaluation of predator-prey relationships in a patchy environment: consequences for habitat use patterns in minnows, Ecology, 62 (1982), 307-313.
doi: 10.2307/1938947. |
[7] |
J. F. Fujita, M. S. and R. V. Bowen, Ant foraging behavior: Ambient temperature influences
prey selection, Behavioral Ecology and Sociobiology, 15 65–68. |
[8] |
S. Ghosh and S. Bhattacharyya,
A two-patch prey-predator model with food-gathering activity, Journal of Applied Mathematics and Computing, 37 (2011), 497-521.
doi: 10.1007/s12190-010-0446-z. |
[9] |
D. Grünbaum and V. R. Richard,
Black-browed albatrosses foraging on antarctic krill: density-dependence through local enhancement?, Ecology, 84 (2003), 3265-3275.
|
[10] |
I. Hanski,
Metapopulation dynamics, Nature, 396 (1998), 41-49.
|
[11] |
I. Hanski,
Habitat connectivity, habitat continuity, and metapopulations in dynamic landscapes, Oikos, 87 (1999), 209-219.
doi: 10.2307/3546736. |
[12] |
I. Hanski,
Metapopulation Ecology Oxford University Press, Oxford, 1999. |
[13] |
I. Hanski and M. Gilpin,
Metapopulation dynamics: Brief history and conceptual domain, Biological Journal of the Linnean Society, 42 (1991), 3-16.
doi: 10.1016/B978-0-12-284120-0.50004-8. |
[14] |
L. Hansson,
Dispersal and connectivity in metapopulations, Biological Journal of the Linnean Society, 42 (1991), 89-103.
doi: 10.1016/B978-0-12-284120-0.50009-7. |
[15] |
A. Hastings,
Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[16] |
S. Hsu, S. Hubbell and P. Wlatman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[17] |
S. Hsu,
On global stability of a predator-prey system, Mathematical Biosciences, 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[18] |
Y. Huang and O. Diekmann,
Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001), 561-581.
doi: 10.1007/s002850100107. |
[19] |
V. Hutson,
A theorem on average liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.
doi: 10.1007/BF01540776. |
[20] |
R. A. Ims and D. Hjermann, Condition-dependent dispersal, in Dispersal, Oxford University
Press, Oxford, (2001), 203–216. |
[21] |
I. M. J{á}nosi and I. Scheuring,
On the evolution of density dependent dispersal in a spatially structured population model, Journal of Theoretical Biology, 187 (1997), 397-408.
|
[22] |
V. A. Jansen,
Regulation of predator-prey systems through spatial interactions: A possible solution to the paradox of enrichment, Oikos, 74 (1995), 384-390.
doi: 10.2307/3545983. |
[23] |
V. A. Jansen,
The dynamics of two diffusively coupled predator-prey populations, Theoretical Population Biology, 59 (2001), 119-131.
doi: 10.1006/tpbi.2000.1506. |
[24] |
Y. Kang, S. K. Sasmal and K. Messan, A two-patch prey-predator model with dispersal in predators driven by the strength of predation, preprint, arXiv: 1505.03820. |
[25] |
P. Kareiva and G. Odell,
Swarms of predators exhibit ''prey-taxis" if individual predators use area-restricted search, American Naturalist,, 130 (1987), 233-270.
|
[26] |
A. Kiester and M. Slatkin,
A strategy of movement and resource utilization, Theoretical Population Biology, 6 (1974), 1-20.
doi: 10.1016/0040-5809(74)90028-8. |
[27] |
V. Krivan,
Dispersal dynamics: Distribution of lady beetles (coleoptera: Coccinellidae), European Journal of Entomology, 105 (2008), 405-409.
doi: 10.14411/eje.2008.051. |
[28] |
M. Kummel, D. Brown and A. Bruder,
How the aphids got their spots: Predation drives self-organization of aphid colonies in a patchy habitat, Oikos, 122 (2013), 896-906.
doi: 10.1111/j.1600-0706.2012.20805.x. |
[29] |
X. Liu and L. Chen,
Complex dynamics of Holling type Ⅱ Lotka--Volterra predator--prey system with impulsive perturbations on the predator, Chaos, Solitons & Fractals, 16 (2003), 311-320.
doi: 10.1016/S0960-0779(02)00408-3. |
[30] |
Y. Liu,
The Dynamical Behavior of a two Patch Predator-prey Model Honor Thesis, from The College of William and Mary, 2010. |
[31] |
G. P. Markin,
Foraging behavior of the argentine ant in a california citrus grove, Journal of Economic Entomology, 63 (1970), 740-744.
doi: 10.1093/jee/63.3.740. |
[32] |
M. Massot, J. Clobert, P. Lorenzon and J.-M. Rossi,
Condition-dependent dispersal and ontogeny of the dispersal behaviour: An experimental approach, Journal of Animal Ecology, 71 (2002), 253-261.
doi: 10.1046/j.1365-2656.2002.00592.x. |
[33] |
E. Matthysen,
Density-dependent dispersal in birds and mammals, Ecography, 28 (2005), 403-416.
doi: 10.1111/j.0906-7590.2005.04073.x. |
[34] |
T. Namba,
Density-dependent dispersal and spatial distribution of a population, Journal of Theoretical Biology, 86 (1980), 351-363.
doi: 10.1016/0022-5193(80)90011-9. |
[35] |
D. Nguyen-Ngoc, T. Nguyen-Huu and P. Auger,
Effects of fast density dependent dispersal on pre-emptive competition dynamics, Ecological Complexity, 10 (2012), 26-33.
doi: 10.1016/j.ecocom.2011.12.003. |
[36] |
J. Poggiale,
From behavioural to population level: Growth and competition, Mathematical and Computer Modelling, 27 (1998), 41-49.
doi: 10.1016/S0895-7177(98)00004-1. |
[37] |
J. D. Reeve,
Environmental variability, migration, and persistence in host-parasitoid systems, American Naturalist, 132 (1988), 810-836.
doi: 10.1086/284891. |
[38] |
O. Ronce, I. Olivieri, J. Clobert and E. G. Danchin, Perspectives on the study of dispersal evolution, Oxford University Press, Oxford, 2001.
![]() |
[39] |
M. L. Rosenzweig and R. H. MacArthur,
Graphical representation and stability conditions of predator-prey interactions, American Naturalist, 97 (1963), 209-223.
doi: 10.1086/282272. |
[40] |
J. F. Savino and R. A. Stein,
Behavioural interactions between fish predators and their prey: effects of plant density, Animal Behaviour, 37 (1989), 311-321.
doi: 10.1016/0003-3472(89)90120-6. |
[41] |
J. A. Silva, M. L. De Castro and D. A. Justo,
Stability in a metapopulation model with density-dependent dispersal, Bulletin of Mathematical Biology, 63 (2001), 485-505.
doi: 10.1006/bulm.2000.0221. |
[42] |
A. T. Smith and M. M. Peacock,
Conspecific attraction and the determination of metapopulation colonization rates, Conservation Biology, 4 (1990), 320-323.
doi: 10.1111/j.1523-1739.1990.tb00294.x. |
[43] |
J. Stamps,
Conspecific attraction and aggregation in territorial species, American Naturalist, 131 (1988), 329-347.
doi: 10.1086/284793. |
[44] |
Foraging behavior of ants: experiments with two species of myrmecine ants, Behavioral Ecology and Sociobiology, 2 (1977), 147–167. |
[45] |
H. R. Thieme,
Mathematics in Population Biology Princeton University Press, 2003. |










Scenarios | |||||||
LAS | ✗ | Saddle | ✗ | ✗ | Saddle | ✗ | |
LAS | Saddle | Saddle | ✗ | ✗ | Saddle | ✗ | |
✗ | ✗ | Saddle | ✗ | ✗ | LAS | Saddle | |
✗ | ✗ | LAS | Saddle | ✗ | ✗ | ✗ | |
✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
Scenarios | |||||||
LAS | ✗ | Saddle | ✗ | ✗ | Saddle | ✗ | |
LAS | Saddle | Saddle | ✗ | ✗ | Saddle | ✗ | |
✗ | ✗ | Saddle | ✗ | ✗ | LAS | Saddle | |
✗ | ✗ | LAS | Saddle | ✗ | ✗ | ✗ | |
✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
Scenarios | |||||||
E11b | E12b | E21b | E22b | E11, 12b | E21b | E22b | |
s ≤ 0.1 | Saddle | ✗ | Saddle | ✗ | ✗ | Saddle | ✗ |
0.15 ≤ s ≤ 0.45 | Saddle | Saddle | Saddle | ✗ | ✗ | Saddle | ✗ |
0.55 ≤ s ≤ 0.62 | ✗ | ✗ | Saddle | ✗ | ✗ | Saddle | Saddle |
0.68 < s < 0.82 | ✗ | ✗ | LAS | Saddle | ✗ | ✗ | ✗ |
s ≥ 0.82 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
Scenarios | |||||||
E11b | E12b | E21b | E22b | E11, 12b | E21b | E22b | |
s ≤ 0.1 | Saddle | ✗ | Saddle | ✗ | ✗ | Saddle | ✗ |
0.15 ≤ s ≤ 0.45 | Saddle | Saddle | Saddle | ✗ | ✗ | Saddle | ✗ |
0.55 ≤ s ≤ 0.62 | ✗ | ✗ | Saddle | ✗ | ✗ | Saddle | Saddle |
0.68 < s < 0.82 | ✗ | ✗ | LAS | Saddle | ✗ | ✗ | ✗ |
s ≥ 0.82 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
Scenarios | Existence condition, Local and Global stability of Model (3) | ||
Always exist and always saddle | Always exist and always saddle | Always exist and always saddle | |
Always exist; LAS and GAS if |
Always exist; GAS if |
Always exist; GAS if |
|
Do not exist | One or two exist if |
Exist if |
|
Exist if |
Do not exist | Do not exist | |
Cond. 1: Cond. 2: |
Scenarios | Existence condition, Local and Global stability of Model (3) | ||
Always exist and always saddle | Always exist and always saddle | Always exist and always saddle | |
Always exist; LAS and GAS if |
Always exist; GAS if |
Always exist; GAS if |
|
Do not exist | One or two exist if |
Exist if |
|
Exist if |
Do not exist | Do not exist | |
Cond. 1: Cond. 2: |
Scenarios | ||||||
s ≤ 0.07 | Source | ✗ | ✗ | Saddle | Source | LAS |
0.9 ≤ s ≤ 0.15 | Source | ✗ | ✗ | Saddle | Saddle | LAS |
0.2 ≤ s ≤ 0.43 | Saddle | ✗ | ✗ | LAS | Saddle | Saddle |
0.68 < s < 0.82 | ✗ | ✗ | LAS | ✗ | ✗ | ✗ |
s ≥ 0.82 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
0.83 ≤ s ≤ 0.84 | Saddle | Saddle | LAS | Saddle | ✗ | ✗ |
s ≥ 0.84 | Saddle | Saddle | Saddle | Saddle | ✗ | ✗ |
Scenarios | ||||||
s ≤ 0.07 | Source | ✗ | ✗ | Saddle | Source | LAS |
0.9 ≤ s ≤ 0.15 | Source | ✗ | ✗ | Saddle | Saddle | LAS |
0.2 ≤ s ≤ 0.43 | Saddle | ✗ | ✗ | LAS | Saddle | Saddle |
0.68 < s < 0.82 | ✗ | ✗ | LAS | ✗ | ✗ | ✗ |
s ≥ 0.82 | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |
0.83 ≤ s ≤ 0.84 | Saddle | Saddle | LAS | Saddle | ✗ | ✗ |
s ≥ 0.84 | Saddle | Saddle | Saddle | Saddle | ✗ | ✗ |
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