Textbook Trends in Biomathematics Modeling Optimization and Computational Problems Selected Works From The Biomat Consortium Lectures Moscow 2017 Rubem P Mondaini Ebook All Chapter PDF
Textbook Trends in Biomathematics Modeling Optimization and Computational Problems Selected Works From The Biomat Consortium Lectures Moscow 2017 Rubem P Mondaini Ebook All Chapter PDF
Textbook Trends in Biomathematics Modeling Optimization and Computational Problems Selected Works From The Biomat Consortium Lectures Moscow 2017 Rubem P Mondaini Ebook All Chapter PDF
https://textbookfull.com/product/applications-of-intelligent-
optimization-in-biology-and-medicine-current-trends-and-open-
problems-1st-edition-aboul-ella-hassanien/
https://textbookfull.com/product/lectures-on-convex-optimization-
yurii-nesterov/
https://textbookfull.com/product/selected-works-dorogovtsev/
https://textbookfull.com/product/advances-in-structural-
engineering-optimization-emerging-trends-in-structural-
optimization-sinan-melih-nigdeli/
Rubem P. Mondaini Editor
Trends in
Biomathematics:
Modeling, Optimization
and Computational
Problems
Selected works from the BIOMAT
Consortium Lectures, Moscow 2017
Trends in Biomathematics: Modeling, Optimization
and Computational Problems
Rubem P. Mondaini
Editor
Trends in Biomathematics:
Modeling, Optimization
and Computational Problems
Selected works from the BIOMAT
Consortium Lectures, Moscow 2017
123
Editor
Rubem P. Mondaini
President, BIOMAT Consortium – International
Institute for Interdisciplinary Sciences
Rio de Janeiro, Brazil
Federal University of Rio de Janeiro
Rio de Janeiro, Brazil
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The present book is a collection of papers which have been accepted for publication
after a peer review evaluation by the Editorial Board of the BIOMAT Consortium
(http://www.biomat.org) and ad-hoc international referees. These papers have been
presented at the technical sessions of the BIOMAT 2017 International Symposium,
the 17th Symposium of the BIOMAT Series which was held at the Institute of
Numerical Mathematics of the Russian Academy of Sciences–Russia, from 30th
October to 03rd November 2017. On behalf of the BIOMAT Consortium, we thank
the Director of the Institute, Prof. Eugene Tyrtyshnikov, and the Co-Chairs of the
BIOMAT 2017 Local Organizing Committee, Prof. Yuri Vassilevski and Prof. Vitaly
Volpert, for their technical expertise in following the guidelines and fine tradition of
the BIOMAT Consortium for preserving the excellency of the BIOMAT Sympo-
sium Series on this first BIOMAT Conference in Russia. Research collaborators,
Ph.D. Students and Secretaries of the Institute like Alexander Danilov, Tatiana
Dobroserdova, Konstantin Novikov, Roman Pryamonosov, Nina Gorodnova and
Anna Zagumennykh have done their best to provide all local technical facilities
to help the speakers during the scientific sessions as well as to follow the Scientific
Programme of the BIOMAT 2017. We are so much indebted for their invaluable help
since the Opening Session on Monday morning to the Closing Session on Friday
evening.
Financial support in terms of accommodation, lunches and coffee-breaks has
been provided by the Institute of Numerical Mathematics, the International Union
of Biological Sciences, the Interdisciplinary Scientific Center Jean-Victor Poncelet,
Russian Foundation for Basic Research, Parseco Foundation, and the Federal
Agency for Scientific Organizations. We are also indebted for the special offer of
accommodation of the Keynote Speakers and the staff of the BIOMAT Consortium
in the hotel of the Steklov Mathematical Institute, Russian Academy of Sciences.
The BIOMAT Consortium has succeeded once more in its fundamental mission
of enhancing the interdisciplinary scientific activities of Mathematical and Biolog-
ical Sciences in Developing Countries with the organization of the BIOMAT 2017
International Symposium. Participants from Western and Eastern Europe, Asia,
v
vi Preface
Africa, North and South America had the usual opportunity of exchanging scientific
feedback of their research fields with their colleagues from Russian Federation and
delegates coming from other 13 countries: Serbia, France, Brazil, Cuba, China P.R.,
Morocco, India, UK, USA, Pakistan, Portugal, Hungary, Italy.
The Editor of the book and President of the BIOMAT Consortium is very glad
for the collaboration and critical support of his wife Carmem Lucia on the editorial
work, from the reception of submitted papers for the peer review procedure of
BIOMAT Consortium Editorial Board to the ultimate publication of the Scientific
Programme. He also thanks his research student Simão C. de Albuquerque Neto
from the Federal University of Rio de Janeiro for his computational skills and
technical expertise with LaTeX versions.
vii
viii Editorial Board of the BIOMAT Consortium
ix
x Contents
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Evolution of Spatial Patterns in
Host-Parasitoid Metapopulation
1 Introduction
B. K. Singh
The University of Notre Dame, Department of Biological Sciences, Notre Dame, IN, USA
S. Sinha ()
Department of Biological Sciences, Indian Institute of Science Education and Research Mohali,
Mohali, India
e-mail: ssinha@iisermohali.ac.in
where F (·) = H μ(1 − H )e−βP and G(·) = H (1 − e−βP ).
The parameters, μ and β, are the host growth rate and the parasitoid attack
rate, respectively. H and P are the post-dispersal host and parasitoid population
densities at any site s ≡ (x, y) ∈ L ≡ (l × l), where L is a square lattice of
l 2 number of habitat patches. Variants of this model have been studied by several
authors [13, 14] while investigating different questions.
Evolution of Spatial Patterns in Host-Parasitoid Metapopulation 3
d1
8
H = (1 − d1 )Ht (s) + Ht (j )
8
j =1
8
P = (1 − d2 )Pt (s) + d2
j
Pt (j )δt (s). (2)
j =1
where d1 and d2 are the host and parasitoid dispersal coefficients, respectively. The
dispersal of the host populations from any patch is independent of the population
level of the destination patches, it simply depends on its own population size. The
parasitoid dispersal is assumed to be dependent on both the host and the parasitoid
j
densities of the neighbouring patches [13, 14]. The term δt denotes the proportion
of dispersing parasitoid populations from the neighbouring sites (j ’s) to site s. The
j
functional form of δt is given by:
j
η
j Ht (s)
δt (s) = CN 8 j
,
i=1 Ht (i)
3 Heterogeneous Landscapes
4 Results
It was shown earlier [20] that in a homogeneous metapopulation, where all sites
are occupied by H and P subpopulations with same life history parameters (r and
β), the spatial dynamics shows complete synchrony in the majority (>90%) of
simulations done with different initial conditions. In very few cases, the lattice
metapopulations showed slow evolution of large scale spatial patterns (e.g., spirals)
in their population distribution (as shown in Fig. 1a). These spatial structures,
when simulated long enough (>10,000 generations), slowly disappeared at the
boundaries. This statistics changes completely when low levels of heterogeneity,
in terms of randomly distributed 5% vacant sites, are present in the landscape
(Type Ia heterogeneity). The presence of such heterogeneity in the landscape resists
the process of spatial synchronization completely, and all lattice metapopulations
(with different initial population distributions) showed spiral patterns in population
distribution (Fig. 1b). However, these spatial structures in Fig. 1b not only evolved
much faster than those seen in the homogeneous metapopulations (Fig. 1a), the
spiral tips remain pinned to any of the vacant sites and the pattern was stable.
For Type Ib heterogeneity (clusters of 9 sites group of vacant patches distributed
randomly), the metapopulation showed similar spatial patterns, i.e., complete spatial
asynchrony with broken spirals. Here again the time taken for the pattern to evolve
was faster than case of Type Ia, and the pattern was stable. It is obvious that
low levels of landscape heterogeneity (Type Ia,b) lead to different numbers of
neighbours to HP subpopulations in habitable patches. For Type Ia heterogeneity,
depending on the distribution of these vacant sites, a subpopulation in a habitable
6 B. K. Singh and S. Sinha
site can in one extreme be solitary (i.e., all neighbours are vacant sites) or have all 8
neighbours. In general the number of habitable neighbours would be 1, 2, 3 to 7. This
local asymmetry can introduce nucleation of spatial patterns once they arise and
stabilize them as seen in Type Ia. The clustered pattern of landscape heterogeneity
(Type Ib) introduces smaller numbers of larger vacant regions. This has stronger
effect on breaking up the different parts of the spiral into smaller dynamical regions
thereby introducing irregular spatial patterns.
Lattices with Type Ic of patterned landscape heterogeneity was studied for
two cases—the vacant patches creating (1) an impermeable barrier and dividing
the metapopulation into two disconnected domains of homogeneous landscapes,
and, (2) a barrier of vacant line of sites with one or more passages through
which limited population dispersal can occur between the domains as shown
in Fig. 1d–f. In case of the impermeable barrier (Fig. 1d), the spatial patterns
formed in the two separate metapopulations are similar to the ones as observed
for homogeneous landscapes (Fig. 1a). In general spiral patterns evolve in these
domains, but irrespective of the size, that domain which does not contain the spiral
core shows spatial synchronization. Figure 1e and f show that the presence of one or
more passages (i.e., habitable sites having a HP subpopulation that can disperse to
neighbours in both domains) in the barrier prevents spatial synchronization. These
connecting subpopulations act as continuous sources of dynamic heterogeneity in
space and lead to generation of semi-circular travelling waves that get absorbed at
the boundaries (Fig. 1d). For the case of more than one passages (Fig. 1f), the waves
created at the different sources interfere with each other, and create complex wave
patterns depending upon the domain size. For larger domains, the wave front can
evolve to appear as if it was created by a single passage. However, if the domain
size is small, each wavefront does not get sufficient space and time to develop into a
single wave front. A detailed study of such barriers with breaks in real landscapes,
in terms of major geographical features and transport links, was shown in spatial
spread of the 2001 UK FMD epidemic [27].
It was shown earlier [20, 24] that the impact of demographic variability on the
dynamics of the spatial HP system is similar to the landscape heterogeneity,
i.e., it induces increasing number of cases of spatial asynchrony as parametric
heterogeneity among the subpopulations increase. The left plot in Fig. 2a for β = 4
is similar to Fig. 1a, and is an example of the case where the metapopulation
in a homogeneous landscape with no demographic heterogeneity shows spiral-
like spatial patterns. The right plot in Fig. 2a for β = 5 (chaotic dynamics in
single HP population) shows asynchrony with irregular pattern in population size
distributions for majority of the cases. Here we show that the spatial patterns of
the population abundance not only respond differently to different types of the
demographic heterogeneity, but also to the predominant intrinsic dynamics of the
Evolution of Spatial Patterns in Host-Parasitoid Metapopulation 7
HP system. The left plot in Fig. 2b shows the spatial pattern in the metapopulations
with Type IIa heterogeneity (i.e., 5% randomly selected sites having parasitoid
populations with β = 5) while the rest have β = 4. The right plot shows Type
IIb heterogeneity (i.e., 5% randomly selected sites have parasitoid populations with
β = 4, while the rest have β = 5). Both plots in Fig. 2b appear more incoherent
and take longer time to reach any coherent patterns (such as spirals) than what was
needed for demographically homogeneous case (Fig. 2a). It was observed that these
spatial patterns were unstable even after 5×104 generations for small lattices as used
in this study.
The Types IIc and IId of demographic heterogeneity in metapopulations
represent cases where 4% of the total sites (i.e. 100 sites) form a single sub-lattice of
(10 × 10) sites, where the parasitoid populations have attack rate β = 5, while the
rest of the sites have β = 4 (Type IIc) and vice versa (Type IId). The spatial
patterns are shown in Fig. 2c. For Type IIc, the pattern is similar to Type IIa
(Fig. 2b left plot), except that the block of sites with β = 5 can pin the spiral core.
It was observed that if the sub-lattice placed randomly happens to cover the core
of a spiral wave, then spatial synchronization emerges finally. On the other hand,
the spatial pattern for Type IId heterogeneity is quite different. The block of sites
having HP subpopulations with β = 4 showed coherent patterns of spiral waves or
of concentric circular waves with their core fixed at the sub-lattice. The concentric
circular pattern was common (≈80% of cases).
The emergence of wave patterns (spiral or circular) in the case of these two
types of heterogeneity (Type IIc and d) can be explained as an outcome of
interaction between the two types of dynamics exhibited by the HP system for
8 B. K. Singh and S. Sinha
5 Conclusions
have indirect adaptive value [20]. This argument, of course, remains valid till the
loss or destruction of habitat patches remain below a threshold beyond which the
habitat will get transformed into disconnected patches [4], which in turn will lead
to the eventual collapse of the metapopulation due to lack of resources and space.
Although it has been shown that the emerging spiral wave patterns are very
robust to environmental noise [8, 31], one of the intriguing questions that the
ecologists are grappling with is the observation and verification of such spatial
structures in field studies. Needless to say, any direct observation of the spiral wave
patterns in the empirical data is difficult, even though attempts are made towards
that [12, 32, 33]. The two major hurdles to uncover such patterns are—lack of
relevant spatial data, and the long transient period that the space-time dynamics
needs to settle down on any discernible geometrical patterns. This task becomes
more difficult with the frequent anthropogenic activities, recognized as one of most
causal factors for the habitat destructions and fragmentations [21]. The latter and its
consequences are perceived to have challenged the ecologists with a daunting task
of analysing and understanding the responses of various ecosystems to externally
induced perturbations [34].
This work has shown the differences in spatial patterns exhibited by metapopu-
lations for random single site variations and patterned heterogeneities. Getting an
exact parallel of the effect of patterned vacancy on the spatial patterns in natural
populations might be as difficult as the direct confirmation of the spiral waves (or
spatial chaos) in natural ecosystems [12, 32]. However, there are many examples
of human developmental activities that lead to the division of an otherwise single
landscape into many domains separated by the patterned barriers. The presence
of such barriers is seen everywhere [35]. As a specific example one can consider
the construction of the Indira Gandhi Canal (≈550 km long) in the late 1950s in
the north-western districts of Rajasthan, India. The canal with its tributaries runs
through a landscape that used to be a part (11%) of the great Indian desert. Needless
to say, the ecological impact of the canal on the species abundances in the region is
quite severe in terms of the loss of endemic biodiversity, as well as the emergence
of new human diseases, among other things [36].
The existence of low levels of heterogeneity—in number of neighbouring
patches to which migration can occur, or neighbours having different demographic
parameters/traits—essentially creates local anisotropies in an otherwise homoge-
neous space. These act like defects inducing synchronization failure in lattices. A
vacant site can act to pin the core of the spiral (like a quenched disorder), thereby
making it robust. Thus spatial asynchrony tends to persist for longer time even in
small lattices. Parametric inhomogeneity induces phase defects through difference
in the local dynamics between neighbouring sites, and induce spatial asynchrony
[37]. For demographically heterogeneous landscapes (Type II), at low levels of
dispersal, phase differences introduced due to difference in intrinsic dynamics,
even at few patches, will keep propagating through several hundreds or thousands
of generations before the metapopulation finally gets spatially synchronized [38].
The ecological implication of the above observation throws up the question as to
if the importance of transient spatial dynamics ever reduces in relevant ecological
10 B. K. Singh and S. Sinha
timescale. Then any effort to look for or anticipate stable spiral wave patterns in
species abundances may prove to be futile. Such instances of long term transient
dynamics were reported in other studies of spatially structured population models,
and the cautionary conclusions were drawn that the transient dynamics should
receive greater attention in order to fully understand the system’s functional
complexity than what it currently does [39].
In this study we have considered the presence of different types of isolated and
patterned heterogeneities (migration barriers or genotypic coexistence) one at a
time. However, in nature they are more likely to overlap. Identifying the natural
barriers, predicting the interaction among physical landscapes, species dispersal,
and gene flow, and using the information to manage ecological and epidemiological
functions are important problems [40]. Needless to say, the spatial patterns in
species populations will be exceedingly difficult to comprehend in the presence
of the environmental heterogeneities that have such combinatorial forms. But
ecologists continuously endeavour to find evidence(s) of stable patterns (spiral wave
patterns) in natural populations [41, 42].
Acknowledgements BKS thanks Prof (Dr.) Paulien Hogeweg for stimulating discussions. SS
thanks J C Bose Fellowship (DST), CPSDE (MHRD-CoE) and Indian National Science Academy
for funding.
References
1. D. Mollison, Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39,
283–326 (1977)
2. M.C. Cross, P.C. Hohenberg, Pattern formations outside of equilibrium. Rev. Mod. Phys. 65,
851–1112 (1993)
3. I. Hanski, Metapopulation Ecology (Oxford University Press, Oxford, 1999)
4. R.V. Sole, B. Goodwin, Signs of Life: How Complexity Pervades Biology (Basic Books, New
York, 2000)
5. S.A. Levin, Dispersion and population interactions. Am. Nat. 108, 207–228 (1974)
6. A. Hastings, Spatial heterogeneity and ecological models. Ecology 71, 426–428 (1990)
7. M.P. Hassell, H.M. Comins, R.M. May, Spatial structure and chaos in insect population
dynamics. Nature 353, 255–258 (1991)
8. H.N. Comins, M.P. Hassell, R.M. May, The spatial dynamics of host-parasitoid systems. J.
Anim. Ecol. 61, 735–748 (1992)
9. R.V. Sole, J. Valls, J. Bascomte, Spiral waves, chaos and multiple attractors in lattice models
of interacting populations. Phys. Lett. A 166, 123–128 (1992)
10. A. Hastings, Complex interactions between dispersal and dynamics: lessons from coupled
logistic equations. Ecology 74, 1362–1372 (1993)
11. M.P. Hassell, H.M. Comins, R.M. May, Species coexistent and self-organising spatial dynam-
ics. Nature 370, 290–292 (1994)
12. M.P. Hassell, Host-parasitoid population dynamics. J. Anim. Ecol. 69, 543–566 (2000)
13. P. Rohani, O. Miramontes, Host-parasitoid metapopulations, the consequences of parasitoid
aggregation on spatial dynamics and searching efficiency. Proc. R. Soc. (Lond.) B 260,
335–342 (1995)
Evolution of Spatial Patterns in Host-Parasitoid Metapopulation 11
14. N.J. Savill, P. Rohani, P. Hogeweg, Self-reinforcing spatial patterns enslave evolution in a host-
parasitoid system. J. Theor. Biol. 188, 11–20 (1997)
15. A. Moilanen, I. Hanski, Metapopulation dynamics, effects of habitat quality and landscape
structure. Ecology 79, 2503–2515 (1998)
16. A.J. Nicholson, V.A. Bailey, The balance of animal populations. Part 1. Proc. Zool. Soc. Lond.
3, 551–598 (1935)
17. R.D. Holt, M.P. Hassell, Environmental heterogeneity and the stability of host-parasitoid
interactions. J. Anim. Ecol. 62, 89–100 (1993)
18. R.M. May, Host-parasitoid systems in patchy environments: a phenomenological model. J.
Anim. Ecol. 47, 833–843 (1978)
19. J.D. Reeve, Environmental variability, migration, and persistence in host-parasitoid systems.
Am. Nat. 132, 810–836 (1988)
20. B.K. Singh, J.S. Rao, R. Ramaswamy, S. Sinha, The role of heterogeneity on the spatiotemporal
dynamics of host-parasite metapopulation. Ecol. Model. 180, 435–443 (2004)
21. World Bank, Sustainable Development in a Dynamic World: Transforming Institutions,
Growth, and Quality of Life, Chap. 1 (The World Bank, Washington, 2003)
22. P.L. Chesson, W.W. Murdoch, Aggregation of risk: relationships among host-parasitoid
models. Am. Nat. 127, 696–715 (1986)
23. T.H. Jones, M.P. Hassell, S.W. Pacala, Spatial heterogeneity and population dynamics of a
host-parasitoid system. J. Anim. Ecol. 62, 251–262 (1993)
24. B.K. Singh, Modelling infectious diseases: spatiotemporal dynamics and control. Ph.D. thesis,
Jawaharlal Nehru University, New Delhi, 2001
25. B.K. Ehlers, C.F. Damgaard, F. Laroche, Intraspecific genetic variation and species coexistence
in plant communities. Biol. Lett. 12, 20150853 (2016). http://dx.doi.org/10.1098/rsbl.2015.
0853
26. S. Cadel-Six, C. Peyraud-Thomas, L. Brient, N.T. Marsac, R. Rippka, A. Mejean, Different
genotypes of anatoxin-producing cyanobacteria coexist in the Tarn river, France. Appl.
Environ. Microbiol. 73(23), 7605–7614 (2007). https://doi.org/10.1128/AEM.01225-07
27. N.J. Savill, D.J. Shaw, R. Deardon, M.J. Tildesley, M.J. Keeling, M.E.J. Woolhouse, S.P.
Brooks, B.T. Grenfell, Topographic determinants of foot and mouth disease transmission in
the UK 2001 epidemic. BMC Vet. Res. 2, 3 (2006). https://doi.org/10.1186/1746-6148-2-3
28. P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)
29. A.H. Hirzel, R.M. Nisbet, W.M. Murdoch, Host-parasitoid spatial dynamics in heterogeneous
landscapes. Oikos 116, 2082–2096 (2007). https://doi.org/10.1111/j.2007.0030-1299.15976.x
30. R.V. Sole, J. Bascompte, J. Valls, Nonequilibrium dynamics in lattice ecosystems: chaotic
stability and dissipative structures. Chaos 2, 387–395 (1992)
31. G.D. Ruxton, P. Rohani, The consequences of stochasticity for self-organized spatial dynamics,
persistence and coexistence in spatially extended host-parasitoid communities. Proc. R. Soc.
Lond. B 263, 625–631 (1996)
32. H.N. Comins, M.P. Hassell, Persistence of multispecies host-parasitoid interactions in spatially
distributed models with local dispersal. J. Theor. Biol. 183, 19–28 (1996)
33. O.N. Bjornstad, J. Bascompte, Synchrony and second-order spatial correlation in host-
parasitoid systems. J. Anim. Ecol. 70, 924–933 (2001)
34. P.M. Kareiva, J.G. Kingsolver, R.B. Huey (eds.), Biotic Interactions and Global Change
(Sinauer Associates, Sunderland, 1993)
35. M.P. Hassell, H.C.J. Godfray, H.N. Comins, Effects of global change on the dynamics of insect
host-parasitoid interactions, in Biotic Interactions and Global Change, ed. by P.M. Kareiva,
J.G. Kingsolver, R.B. Huey (Sinauer Associates, Sunderland, 1993)
36. I. Prakash, Biological invasion and loss of endemic biodiversity in the Thar desert. Resonance
6, 76–85 (2001)
37. A. Holden, M. Markus, H.G. Othmer (eds.), Non-linear Wave Processes in Excitable Media
(Plenum Press, New York, 1991)
38. B. Blasius, A. Hupert, L. Stone, Complex dynamics and phase synchronization in spatially
extended ecological systems. Nature 399, 354–359 (1999)
12 B. K. Singh and S. Sinha
1 Introduction
systems) generation of the so-called dark states (or dark-state polaritons) known in
quantum optics [8, 9] is possible. We conjecture that these dark states are related to
the effect of quantum photosynthesis.
The scheme of experiments on quantum photosynthesis is as follows: first,
one prepares a photosynthetic system and applies a laser field to the system (the
frequency of the laser should be in resonance to transitions in the photosynthetic
complex); second, one switches off the laser for short time; third, one applies
the same laser field to the system and observes the photonic echo. Our aim is
to reproduce this scheme using our model of dissipative dynamics in degenerate
system.
Dynamics of the system density matrix in the model under consideration is
described by a sum of three generators—the photonic generator, the phononic
generator and the generator of absorption of excitons. We consider the following
scheme of manipulation of quantum states of the system.
1. First, we compute the stationary state of the system taking into account only the
photonic generator and neglecting the effects of transport and absorption (this can
be considered as a strong light approximation). This state will contain quantum
coherencies (off-diagonal part of the density matrix).
2. Second, we investigate joint action of generators of transport and absorption on
the state prepared at the previous step. The observation is that a part of coherences
will survive due to the effect of dark states.
3. Third, we apply the laser field again and show that the interaction of the prepared
state with the laser field will be non-trivial, which gives the possibility of
photonic echo for this state.
In non-degenerate case (when energy level ε2 is non-degenerate) the considered
in this paper non-equilibrium quantum system was discussed in [10] and the flow
of excitons in the system was computed (earlier stochastic limit of nonequilibrium
three level quantum open system was considered in [11]). Open quantum systems
coupled to several reservoirs were considered in many papers, in particular in [12].
In paper [13] effects of degeneracy in exciton transport were investigated by the
stochastic limit method and it was shown that it is possible to achieve quantum
amplification of exciton transport (the supertransport effect). For discussion of
supertransport in quantum photosynthesis, see also [14–17]. The possibility of
excitation of non-decaying dark states in a degenerate system can be considered
as a side effect of the supertransport.
In [18] dark states in photosynthetic systems were studied experimentally. In
[19] dark states in photosynthesis were also considered but dark states in this paper
were considered in a different way compared to the approach of present paper, in
particular in this case the effect considered in the present paper does not take place.
In [5] application of quantum methods to computations and biology was dis-
cussed. Photosynthetic complex is an example of complex physical system, see [20]
for the review of ultrametric approach to complex systems in physics and biology.
The exposition of the present paper is as follows. In Sect. 2 the Hamiltonian
of light-harvesting system interacting with three reservoirs (photons, phonons and
Dark States in Quantum Photosynthesis 15
Let us consider the Hilbert space HS = CN +1 and use the Dirac notations. Let
{|0, . . . , |N } denote an orthonormal basis in HS .
We consider a system with three energy levels ε0 < ε1 < ε2 , where the upper
level is degenerate, with the Hamiltonian (operator in HS )
N
HS = ε0 |00| + ε1 |11| + ε2 |j j |. (1)
j =2
ε2 −ε0 ), transitions between the levels ε2 and ε1 (transport of excitons to the reaction
center) are related to interaction with phonons (described by the Bose field with the
−1
temperature βph = 300 K), and transitions between the levels ε1 and ε0 (absorption
of excitons in the reaction center) are described by interaction with the sink reservoir
in the Fock state (i.e. reservoir with the zero temperature).
Thus we have three reservoirs described by Hamiltonians of quantum Bose fields
Hem (light, or electromagnetic field), Hph (phonons, or vibrations of protein matrix),
Hsink (sink, or absorption of excitons in the reaction center), each of the reservoir
Hamiltonians has the form
HR = ωR (k)aR∗ (k)aR (k)dk, (2)
R3
here aR∗ (k), aR (k) are creation and annihilation operators, [aR (k), aR∗ (q)] = δ(k −
q), and the index R = em, ph, sink enumerates the reservoirs, ωR is the dispersion
of the Bose field aR (some nonnegative function). Each of the reservoirs is in a mean
zero Gaussian state with the quadratic correlator
Here NR (k) (number of the field quanta with wave number k) for the temperature
state is equal to
1
NR (k) = (3)
eβR ωR (k) −1
here function gem (k) is the form-factor of interaction with the field, and the bright
photonic vector χ belongs to the level with energy ε2 , i.e. HS |χ = ε2 |χ .
Transport of excitons to the reaction center is related to interaction of the system
with phonons
HI,ph = Aph |ψ1| + A∗ph |1ψ|, A∗ph = ∗
gph (k)aph (k)dk, (6)
R3
For investigation of the model we will use the method of quantum stochastic limit
[6, 7]. In this limit dynamics of reduced density matrix of a system interacting
with reservoir is generated by quantum dissipative operator in the Lindblad form,
see formulae (10)–(12) below. This approach allows to take into account quantum
decoherence, dissipation, transport and other thermodynamic phenomena. Quantum
dynamics in the stochastic limit in presence of a laser field was discussed in [22].
Properties of antibunched light are discussed, for instance, in [23].
For the considered model with three reservoirs the dynamics will be generated
by a sum of three generators (photonic, phononic and sink)
d
ρ(t) = θem + i[·, Heff ] + θph + θsink (ρ(t)). (8)
dt
Creation and annihilation of excitons are described by the photonic generator
equal to a sum of the dissipative Lindblad term θem and the term i[·, Heff ] related to
interaction with a coherent field [6, 7]:
Lem = θem + i[·, Heff ], Heff = s(|χ 0| + |0χ |), s ∈ R. (9)
− 1 −
θem (ρ) = χ 2
2γre,em χ |ρ|χ|00| − {ρ, |χ χ |} − iγim,em [ρ, |χ χ |]
2
+ 1 +
+ 2γre,em 0|ρ|0|χ χ | − {ρ, |00|} + iγim,em [ρ, |00|] .
2
(10)
The parameter s is the amplitude of the laser field. Here the normed bright photonic
vector has the form
|χ
|χ = .
χ
− 1 −
θph (ρ) = ψ 2
2γre,ph ψ|ρ|ψ|11| − {ρ, |ψψ|} − iγim,ph [ρ, |ψψ|]
2
+ 1 +
+ 2γre,ph 1|ρ|1|ψψ| − {ρ, |11|} + iγim,ph [ρ, |11|] .
2
(11)
Let us note that for large ψ the transport of excitons will be amplified
(the generator is multiplied by ψ 2 ). This corresponds to the supertransport
phenomenon. The maximal possible amplification is equal to the degeneracy of the
upper level.
Absorption of excitons is described by the sink generator
− 1 −
θsink (ρ) = 2γre,sink 1|ρ|1|00| − {ρ, |11|} − iγim,sink [ρ, |11|]. (12)
2
The constants γ have the form (where P. is the principal part, or Cauchy principal
value, generalized function)
+
γre,R =π |gR (k)|2 δ(ωR (k) − ωR )NR (k)dk, (13)
−
γre,R =π |gR (k)|2 δ(ωR (k) − ωR )(NR (k) + 1)dk, (14)
+ 1
γim,R =− |gR (k)|2 P. NR (k)dk, (15)
ωR (k) − ωR
− 1
γim,R =− |gR (k)|2 P. (NR (k) + 1)dk, (16)
ωR (k) − ωR
A few days after the great festival Major S⸺, who was going
down country, kindly allowed me to accompany him as far as
Hamadan. We started one afternoon, doing the two first stages by
sunset, and stopping at the post-house at Karneabad.
The weather was fine, the roads and horses good. I had by this
time learnt to ride by balance only, and acquired the art of remaining
in an upright position on my steed whenever he suddenly dropped as
if shot, instead of going over his head by the force of momentum.
The Major had a few tinned provisions, which it had been impossible
to get in a place like Tiflis, and with a roast fowl or two our
commissariat was well provided. The intense cold was over, and I
was glad to use my goggles to protect my eyes during the middle of
the day. We also never started before the light was good, which
made an immense difference in our comfort.
I had invested in a native bridle, the severe bit of which enabled
me thoroughly to control my horses, and, being the one they were
used to, did not keep them in the perpetual state of fret that the
European bit did. My saddle-bags, too, were well packed and exactly
the same weight, so that I never had to get down to put them level,
and they never annoyed the horse.
I had my rugs, four in number, and the same size, sewn together
down one side and at the bottom, so that whichever side I might
have to the draught, and of this there is always plenty, I could have
one blanket under me, three over me, and the sewn edge to the
wind, while, as the bottom was sewn up, the blankets could never
shift, and the open side could be always kept to the wall. This
arrangement, an original one, I have never altered, for in hot
weather, by lying on say three blankets, one only was over me.
There is, however, one thing that I soon found out in travelling. To
thoroughly rest oneself it is needful to, firstly, undress and wear a
night-jacket and pyjamas; and, secondly, to sleep in a sheet. The
addition to one’s comfort is immense, particularly in warm weather,
while the extra weight of a sheet is not worth considering. An air
pillow, too, is a great luxury.
I have been in the habit of no longer using a waterproof sheet to
keep my blankets dry, but of rolling them tightly up, and then
strapping them and cramming them into an india-rubber soldier’s
hold-all, which ensures a dry bed, and straps handily to the saddle.
This hold-all was the cause of a rather amusing adventure.
On coming home once on leave, in a great hurry, I had left Persia
with only my hold-all, having given my saddle-bags and road kit to
my servant. I had come direct from Tzaritzin on the Volga to
Boulogne without stopping, but had to wait some hours on the tidal
boat before she started. I stepped on board and asked one of the
men where the steward was.
“Oh, he ain’t aboard yet, mate.”
“Can you get me a wash?”
“Come along a’ me, mate.” The man took me down to what
seemed the fo’cassel, and placed a bucket of water before me.
I said, “Come, is this the accommodation you give your first-class
passengers?”
The man roared with laughter.
“No yer doant, mate, no yer doant. I never seed no first-class
passengers with luggage like that,” pointing to my hold-all; and it was
only on producing my coupon book, that the man could be
persuaded I was not a deck passenger, and to take me to the saloon
aft.
As I was covered with coal dust, and generally grimy—the
opportunities for washing being then not what they are now in Russia
and Germany—the hold-all had made the man sure that I was an
impostor.
We came in the afternoon of the third day into Hamadan, having
done the stages in fair time. The journey was without incident, save
that a string of antelopes crossed the road in broad daylight some
ten yards ahead of us. As they appeared so suddenly, we neither of
us thought of using our revolvers. Hamadan looked pretty as we
entered it, and was surrounded by apparently interminable gardens.
On turning a corner we came upon Captain Pierson, under whom I
was to serve, and of whose division I was in medical charge. He had
ridden out to meet us.
In the early days of the Persian Telegraph it was usual to ride out
with the departing, and to do the same to meet the coming guest.
This is the Persian custom of the “istikhbal,” or ceremonious riding
out to meet the new arrival; being a very important ceremony,
regulated by hard-and-fast rules: such as that the greater the
personage, the further must the welcomer travel; while the lesser the
welcomer, the further must he go. Thus, in the case of a new
governor of Shiraz, the king’s son, the big men rode out three
stages, the ex-governor one, while some actually went as far as
Abadeh, or seven days’ journey; but these were mostly merchants or
small people.
Great fuss and parade is made, the condition of the incomer being
denoted by the grandeur of this “istikhbal,” or procession of
welcome. In the case of official personages, soldiers, both horse and
foot, go out; led-horses also are sent simply for show, splendidly
caparisoned with Cashmere shawls or embroidered housings on the
saddles. And it is found necessary, in the case of the arrival of
ambassadors or envoys, such as that of Sir F. Goldsmid (when on
the duty of the definition of the Seistan boundary), to stipulate that a
proper istikhbal shall be sent out prior to the commissioners entering
a large town.
There is another ceremony, that of the “badraghah,” or riding out
with the departing guest. This, however, is not so formal, and is less
an act of ceremony than one of friendship; however, it is a
compliment that in both cases is much appreciated, especially when
shown by a European to a native.
Latterly the Europeans have almost given up this riding out, which
practically is a great nuisance to those riding at an unusual or
uncomfortable time, perhaps in the sun, and when the arrival of the
guest is very uncertain; it is, too, very annoying, when tired with a
rapid chupper, and having ridden many hours on end, to be put on a
very lively horse, ready to jump out of his skin with condition, and to
pull one’s arms off.
As we had got in sooner than was expected, and were only some
mile from Pierson’s house, we did not change our horses for the
fresh ones provided by him, and after many turns and twists between
high mud walls, we came to the house, and here my travels ended
for the time.
The courtyard was some twenty yards by thirty wide. A hauz or
tank ran the entire length, filled by a constant stream of running
water, and on either side of it was a long flower-bed sunk in the
stone pavement, about the same depth below it as the hauz was
elevated above.
On a level with the ground in the basement were the cellars and
servants’ quarters, and above this a platform ten feet from the
ground, some four yards broad, which extended the whole width of
the courtyard. This was covered by an enormous structure,
consisting of a roof some six feet thick, being painted wood mudded
over a yard deep; and then under it a hollow air-chamber, supported
on three huge wooden octagonal columns, likewise painted in red,
blue, and yellow. Behind and beneath this talár, or verandah, which
was some thirty feet from floor to ceiling, was a central room
(orūssēe), elaborately painted and gilt in the vilest taste, with a huge
window (which could be kept wide open in hot weather) of coloured
glass, in small panes four inches by seven. This was the dining and
reception room.
On either side of this orūssēe, and having the talár still in front of
them, was a smaller apartment. One was Pierson’s bedroom, the
other mine. Thus in front of the three rooms was a covered platform,
four yards by twenty. On this during the summer, save when the sun
was on it, we lived, and when the sun was high the rooms were kept
cool by the talár.
We soon sat down to a sumptuous dinner, and I tasted, for the first
time, Hamadan wine, of which I had heard many and different
opinions. It was a delicious pale, scented, straw-coloured wine, like a
light hock; rather too sweet, but apparently of no great strength. I
soon found, however, that in the latter idea I was much in error, for it
was a wine that went straight to the head, and remained there.
Delicious as it is, the fact of its newness—and it often will not
keep, a second summer generally turning it sour if in bottle—makes
it objectionable, for though it is light and delightful, especially when
iced, a headache surely follows even a third glass.
The natives, we found out in after years, are able to keep it in bulk,
and then the tendency to give an after headache goes away, but so
does the delicious flavour. In winter so cold is Hamadan, that the
wine, which is kept in huge jars holding two hundred maunds (or
eight hundred bottles), or even more, sunk half their depth in the
ground, has to be kept from freezing by making a hot-bed of
fermenting horse-dung around the upper part of these jars, and often
these means fail; for I have myself been present when blocks of
frozen wine have been chopped out of the jars for drinking; these
plans of storing wine only refer to Hamadan: in other Persian towns
the wine, as soon as it is cleared, is placed in carboys, holding from
six to twenty-four bottles.
It is sold in Hamadan in baghallis, or native bottles, holding about
a pint and a half. They are of the very thinnest glass, and very fragile
when empty. One of these bottlefuls costs about fourpence—at least
it did when I was in Hamadan in 1869.
In a couple of days Major S⸺ left on his way to Baghdad, and
Pierson insisted on my remaining his guest, which I was only too
glad to do, till I could get servants, etc. of my own.
The first thing, however, was to buy a horse, as I could not draw
my horse allowance from Government till I had really a horse of my
own, and the three pounds a month was, considering the smallness
of my pay, a consideration. Of course at that time I knew nothing
about horses, and was fortunate in having the advice of Pierson. As
soon as it was given out that I wanted horses there was a permanent
levée at our quarters of all the owners of the lame, the halt and the
blind, and their animals. These men, however, were all sent to the
right-about by Pierson, and at last a dealer came with four likely
young horses; these were examined and pronounced sound. On
their price being asked, one hundred tomans each was demanded. I
was disappointed, for this was exactly the sum (forty pounds) that I
was prepared to give for two horses. But I was reassured by Pierson,
who made me understand that that was always the price asked for
any beast worth having, and merely meant that the seller did not
mean to take less than one half the amount. I was told, too, that if
one wanted to buy a horse anywhere near its value some weeks
must be taken in the negotiation. The matter ended in Pierson’s
offering the dealer fifty tomans for two of the animals, and the man
leaving our courtyard in simulated indignation, declining even to
notice a bid so ridiculous. However, as Pierson said we had not seen
the last of him, I did not despair.
Next morning, on coming out to breakfast, I saw our horse-dealer
seated with the servants, and as Pierson put it, “They are settling the
amount of commission they are each to get, and this commission
they will have; ten per cent. is legitimate, more is robbery. So all we
have to do is to be very determined; if you can get any two of the
four animals for your limit you will do well, if not you must let them
go.”
Pierson now sent for his head-man and told him that “I was to
have two serviceable horses for forty pounds, and that I should not
pay a penny more; so, as he knew the amount of modakel (profit) he
and the rest of the servants could make, he had better do the best
he could for me, and that he (Pierson) would see that I was not done
as to quality.” The man cast up his eyes and retired.
While we were at breakfast a poor prince, Abu Saif Mirza, came,
and was invited to partake. Pierson told me that he was a very good
fellow indeed, and a grandson of Futteh-Ali Shah,[7] a former Shah
of Persia, but from the irregularity with which his very small pension
was paid, he had to live almost by his gun, and chance meals, such
as the present.
Of course I could not understand what he said, but he fully entered
into the difficulty as to finding me a horse. And as in Persia nothing
can be done without stratagem, he suggested on the spot a means
for bringing the dealer to his senses; it was deep, “deep as the deep
blue sea.” It was simply this: he would exhibit his horse to Pierson
and promise to send it for trial to-morrow, naming a price just about
its value, and “then you will see all will be well.”
No sooner was breakfast over than the prince’s horse was brought
into the courtyard, stripped and examined, and the suggested
arrangement made. As it happened I afterwards bought this very
horse for Pierson to make a wedding present of, but he would have
been more than I could manage at the time, being a spirited beast
and a puller. The Shahzadeh (prince) took his departure, promising
loudly to send his horse round in the morning.
No sooner was he gone than the nazir, or head-servant, presented
himself and delivered to Pierson an oration somewhat of this sort.
“May I represent to the service of the sahib, that it would be very
unwise to purchase the horse of the prince? he is not young” (he
was five years old), “he is gone in the wind” (he was quite sound),
“and his temper is awful; besides this I have reason to know that he
is worthless in every respect” (he was one of the best horses I ever
saw, and I knew him for ten years). “Of course to me it would make a
great difference, for the prince has indeed offered me a handsome
commission” (quite untrue), “while from this poor dealer not a
farthing can be wrung by the servants. No! he would rather die than
pay one farthing. So though the other servants are loath to let a sale
take place to my sahib’s friend, yet I, as an old servant, and looking
for a reward from my sahib for conduct so disinterested, have after
infinite trouble got the dealer to consent to a hundred and fifty
tomans for any two of his four horses.”
“Be off,” was the laconic reply of Pierson. “When we ride to-day, if
the dealer will sell for my price, let the horses be ready and I will see
them and ride them; if not he can go.”
The man sighed and replied: “Ah, I see, sahib, the prince has
laughed at your beard, and persuaded you to buy his worthless
brute. I can’t offer such terms to a respectable man like the dealer,
but I will give the message.”
I now saw the horse-dealer leave the courtyard with the air of an
injured man, and I feared I was as far off a purchase as ever. But
Pierson reassured me. I had plagued him to sell me one of his own
large stud which he wished to reduce, but he declined with a smile,
saying he never sold a horse to a friend unless he was a thorough
judge, and that as I knew nothing about horses he must decline, as I
might repent when too late; and though I pressed him a good deal,
he would not relent. Few men would have lost an opportunity to get
rid of beasts they did not require, but Pierson was a man in a
thousand.
At that time he had eight horses in his stable, all good and all
sound. He had named them after heathen gods, Jupiter, Pluto,
Saturn, Cupid, Hercules, etc. But his pet nag, Apollo—a grey he had
given one hundred and twenty tomans, or fifty pounds, for, an
enormous price in those days in Persia—had a few weeks before
caught his foot in a hole while galloping over turf; horse and man
came down with a crash; Pierson was insensible; and when he came
to himself he found, some four yards off, his favourite lying dead with
his neck broken.
He rode away on his groom’s horse, the man carrying his saddle
and bridle. On getting to the house he sent a gang to bury the poor
beast, but too late, for the villagers had taken off the skin and tail.
Pierson on telling me the story did it so pathetically, that he left off
with wet eyes, and I felt inclined to sob myself.
As we got ready for the afternoon ride, the horse-dealer and his
four horses appeared, and with a sigh he informed Pierson that he
accepted the terms, or nearly so. On getting out of the town the
horses were put through their paces. They were a big grey, with
enormous mane and tail, of not much breed, but in dealer’s
condition, and a well-shaped and strong-looking beast; an iron-grey,
who plunged and shied and was generally vicious, but really the
most valuable of the four; a fourteen-hand pure-bred Arab, with a
huge scar of a spear-wound a foot long on his shoulder, otherwise
perfect, of angelic temper, but small by the side of the Persian
horses, as all Arabs are; his muzzle almost touched his chest as he
arched his neck, and his action was very high, yet easy; he seemed
an aristocrat compared to the rest; his thin and fine mane and tail
were like silk—he, too, was five. A big, coarse, raking chestnut, that
took all the boy who rode him could do to hold him, rising four,
completed the list.
Pierson kindly rode them all, and with considerable fear I did the
same, save the lively grey, which I wisely acknowledged to be too
much for me. The big chestnut bolted with me, but I stuck on. The
other chestnut was all I could wish, fast, paces good, no tricks,
willing—but, then, the scar. I did not wish to buy him on that account,
but Pierson over-ruled me, and I took his advice; he told me that in
Persia a scar was nothing, that I could ride the horse in comfort and
safety, as he had no vices, and that whenever I wished to sell I
should lose very little. The raking chestnut, as a young horse,
Pierson told me was a speculation; he might turn out well, he might
not. And the grey—well, all I could get out of Pierson was, that “he
had a fine mane and tail,” which he certainly had, and that “he was
value, or nearly.” He was not a well-bred animal, and I liked him, I
fear, on account of the mane and tail; but he pulled. All were entire
horses.
Pierson wouldn’t let me buy the iron-grey, had I wanted to, as he
said he was dangerous, even to a good rider.
So the matter ended in my taking the chestnut for five hundred
and fifty kerans and the grey for six hundred and fifty. Pierson said
the prices ought to have been reversed. He was right. I had that
chestnut Arab ten years; he never was sick or sorry, and I never had
to strike or spur him; a pressure of the knee and a shake of the rein
would make him do his utmost. And he was a fast horse; small as he
was he carried my twelve stone comfortably, and as a ladies’ horse
he was perfect, having a beautiful mouth, while he followed like a
dog, and nothing startled him or made him shy. In the stable he was
quiet, save to a new-comer, on whom he always left his mark by a
bite on the neck, and then, having asserted his position, which was
afterwards never disputed, he was always friendly to stable
companions. He never kicked. I gave him away at last, when I left
Persia on leave.
Next morning the “poor prince” called and looked over my
purchases; he approved the chestnut, but shook his head at the
grey, saying he had “ableh,” or leprosy, and that in time he would
break down, pine, and die. The only sign he had was a pink patch
the size of a fourpenny-piece on his black muzzle. “Give him back,”
said the Prince.
“I can’t see anything wrong,” said Pierson. His mane and tail
decided me. I stuck to him, christening him “Salts”—the chestnut I
called “Senna.”
The custom in Persia is that, until a horse has been three nights
fed in the stable of a new master (unless specially stipulated to the
contrary before witnesses of respectability, or in writing) he may be
returned without giving any reason whatever, simply on the
purchaser repenting his bargain; this is often taken advantage of by
the buyer to return the animal in order to lower his price; the
manœuvre seldom succeeds, as the seller is prepared for it.
The European, if awake to his own interest, generally spends the
three days in giving the beast a good “bucketting” over ploughed
land, when, if there be any hidden defect, it comes out, and the
animal can be returned. We did this, but no fault showing itself, I paid
my one hundred and twenty tomans[8] (forty-eight pounds) and
concluded the purchase.
CHAPTER VI.
HAMADAN.