論文 / 著書情報
Article / Book Information
題目(和文)

ロジスティック成長を伴う反応拡散モデルにおける総個体数の最大化

Title(English)

Maximization of the total population in reaction-diffusion models withlogistic growth

著者(和文)

永原健大郎

Author(English)

Kentaro Nagahara

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第11362号,
授与年月日:2020年3月26日,
学位の種別:課程博士,
審査員:柳田 英二,利根川 吉廣,隠居 良行,川平 友規,小野寺 有紹

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第11362号,
Conferred date:2020/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

要約

Type(English)

OutlinePowered by T2R2 (Tokyo Institute Research Repository)

Maximization of the total populationin reaction-diﬀusion models
with logistic growth
Summary

Kentaro Nagahara
Tokyo Institute of Technology

A thesis submitted for the degree ofDoctor of Science
February 2020

Acknowledgment
The author’s earnest appreciation goes to his advisor Professor Eiji Yanagida
who provided helpful comments and suggestions. A part of this thesis is joint workwith him [20, 21]. Special thanks go to Professor Yuan Lou, who also provided
helpful comments. A part of this thesis is based on joint work with him [20]. Special
thanks also go to Assistant Prof. Masataka Shibata whose comments have madean enormous contribution to his work. His heartfelt appreciation goes to coworkers
of Tokyo Tech High school of Science and Technology, department of mathematics,
Ken-ichi Masuda, Hideto Simoe, Kentaro Tago, Kei Okamoto, Takeshi Hayasaka,
Fumihiko Nomoto for understanding and cooperation on this matter. The author
thanks to students of Eiji Yanagida’s laboratory and students of the same graduate
student room for their moral support and warm encouragements. The author would
like to express his deepest gratitude to his family, Miyuki, Shigemi, Masako forwarm encouragement.

2

Abstract
This thesis is concerned with a nonlinear optimization problem that naturally
arises in population biology. We consider the eﬀect of spacial heterogeneity on the
total population of a biological species at a steady state, using a reaction-diﬀusion
model, or multi-patch model with logistic growth. Our objective is to maximize
the total population when resources are distributed in some patches to control the
intrinsic growth rate, but the total amount of resources is limited. It is shown that
global maximizer is of “bang-bang” type under some constraint. Further, global
maximizer can be expressed in a patchy environment for any number of patcheswith suﬃciently small and large diﬀusion constant. To this purpose, we computeasymptotic expansion of the total population by using perturbation theory in areaction diﬀusion model. In the case patchy environment, we use Taylor expansion
and recurrence relation.

3

Summary
This is a summary of the author’s Ph.D thesis [19] at Tokyo Institute of Technology.

Reaction-diﬀusion model with logistic growth
In this thesis, we first consider the boundary value problem for a semilinear elliptic
equation∆u + u(m(x) − u) = 0 in Ω,
(1)
∂u

=0
on ∂Ω.
∂ν
Here, Ω is a bounded domain in Rd with a smooth boundary ∂Ω, ν is an outward
unit normal vector on ∂Ω, and m∫is a bounded Lebesgue measurable function. Ourinterest is in the dependency of Ω udx on the weight m, specifically with relation
to the reaction-diﬀusion logistic equation.
The problem (1) appears as a stationary problem for the following model ofbiological population growth introduced by Skellam [23], which plays an importantrole in studying the eﬀects of dispersal and spatial heterogeneity in population
dynamics:

v = ∆v + v(m(x) − v) in Ω × R+ ,

 t
∂v
(2)
=0
on ∂Ω × R+ , ∂νv(x, 0) ≥ 0, v(x, 0) ̸≡ 0 in Ω,
where v(x, t) represents the density of a species at location x and time t, andv(x, 0) ∈ C(Ω) is the initial data. In regards to biology, Ω is the habitat of a
species and the function m(x) represents the intrinsic growth rate of a species at
location x. If m(x) > 0, then it is favorable to the species, whereas if m(x) < 0,then it is unfavorable. The zero-flux boundary condition in (1) means that noindividuals cross the boundary of the habitat.
The trivial steady state u ≡ 0 corresponds to the extinction of the species. If
the trivial solution is unstable, then the species can survive, that is, there is aunique positive steady state. Conversely, if u ≡ 0 is stable, then the species has
4

no chance to survive, that is, the solution tends to 0 as t → ∞ uniformly on Ω.
Therefore, the stability of the trivial steady state is crucial for the survival of thespecies. We refer to [2, 4, 5, 14, 15] and reference therein for previous works on(2).
Thus, if v ≡ 0 is unstable, the total population satisfies∫∫v(x, t)dx →u(x)dx as t → ∞,ΩΩwhere u is the positive solution of (1). Note that the solution u of (1) depends
only on the function m. Hence we may denote the total population at steady stateas∫J(m) :=udx.
Ω

Our objective is to maximize the total population J(m) under the following
constraints on m:(M1) m(x) ≤ m(x) ≤ m(x) a.e. in Ω,∫
m(x)dx ≤ M,
(M2)Ω(M3) v ≡ 0 is unstable,
where m, m are given bounded and continuous functions in Ω, and M is a given
constant satisfying∫∫
m(x)dx < M <m(x)dx.
ΩΩWe define the feasible set of m by
M := {m ∈ L∞ (Ω) : m(x) satisfies (M1), (M2), and (M3)} ,(3)and assume that M is not empty.
These constrains (M1), (M2), and (M3) are derived from preservation of anendangered species which may not be able to survive without protection. The
constraint (M1) implies that the intrinsic growth rate at location x is given bym(x) if there is no protection, and that the∫ growth rate becomes at most m(x) if
some protection is provided. The integral Ω mdx represents the total amount of
resources for protection, and (M2) implies that the resource is not enough. Though(M3) does not impose constraint on the function m explicitly, it is a necessary andsuﬃcient condition for the existence of a positive solution of (1) [4, 5, 10, 15]. We
also refer to [3, 11, 22] for related work on the existence and stability conditions
for the population.
5

If a function m∗ ∈ M satisfies
J(m∗ ) = max J(m),m∈M

we call m∗ a maximizer (or an optimal control) of J. Ding et al. [6] carried out
numerical simulation in the case where m ≡ 0 and m ≡ 1, and addressed thefollowing conjecture:
Conjecture
A. Assume that weight m ∈ L∞ (Ω) satisfies 0 ≤ m(x) ≤ 1 and∫mdx = M , where 0 < M < |Ω|. Then there exists a measurable set E such thatΩ
the optimal control of J(m) satisfies m(x) = 0 on E and m(x) = 1 on Ω \ E.
Here m ∈ M is said to have a “bang-bang” property if m is expressed asm(x) = m(x)χE + m(x)χΩ\E a.e. for some measurable set E ⊂ Ω, where χAdenotes the characteristic function. In this paper, we show that a similar situationof this conjecture holds true in a more general situation where the upper boundm and lower bound m of (M2) are not constant.
Our main result can be stated as follows:
Theorem 1. m ∈ M is not a local maximizer of J(m) if there exists δ > 0 and anonempty open set Ωδ such that m(x) + δ < m(x) < m(x) − δ almost everywherein Ωδ .
As is well known, if m is Riemann integrable, then m is continuous almost
everywhere. Hence the following corollary follows immediately.
Corollary 2. Suppose that m ∈ M is Riemann integrable. If m is a local maximizer of J(m), then m is of bang-bang type.
When m is not of bang-bang type, the set{x ∈ Ω : m(x) < m(x) < m(x)}has a positive measure, but may not have an interior point. In this case, it is very
diﬃcult to show the bang-bang property of local maximizers.
To show the bang-bang property, it might be useful to consider rearrangement
argument when upper and lower bound of m are constant (see, e.g., [17, 18]).
However, this method is not eﬀective in our case. In this paper, we adopt a
diﬀerent approach, namely, we apply the perturbation method, and carry out
careful analysis on the first and second variations of J when m ∈ M varies.

6

Multi-patch model
Understanding the eﬀect of dispersal in heterogeneous environment on population
dynamics is an important issue in spatial ecology [5]. Generally large diﬀusion
tends to reduce the spatial variations in population distributions, while small diffusion might help organisms adapt to the local environment. In this paper we areinterested in the impact of dispersal upon the total population of a single species
residing in a spatially heterogeneous patchy environment. More specifically, we ask
the following question: Given the total amount of resources, how should we distribute the resources across the habitat in order to maximize the total populationof a species?
To address this question, we consider the following system of ordinary diﬀerential equations for a single species with logistic growth in a patchy environment:

d

∗

 dt vi (t) = vi (mi − vi ) + δ(vi−1 + vi+1 − 2vi ), i ∈ Ω , t ∈ R+ ,(4)v0 (t) = v1 (t), vN +1 (t) = vN (t),t ∈ R+ ,∑i ∈ Ω∗ ,vi (0) ≥ 0, N
i=1 vi (0) > 0,
where N ≥ 2, Ω∗ := {1, 2, . . . , N }, and {mi }i∈Ω∗ ⊂ R is a sequence which satisfies
mi ≥ 0,N∑

mi = m > 0.

(5)

i=1

The problem (4) was first studied by Levins [13], as a multi-patch model for asingle species, where N is the total number of patches and δ > 0 is the diﬀusion
rate. The unknown function vi (t), i ∈ Ω∗ , t ∈ R≥0 := [0, ∞), denotes the numberof individuals in i-th patch at time t. The constant mi , i ∈ Ω∗ , represents theintrinsic growth rate of the species in i-th patch. If mi > 0, then i-th patch isfavorable to the species. The second equation in (4) means that no individuals
cross the boundary of the habitat, so system (4) is closed. The constraint (5)means that the total amount of resources is limited.
Under assumption (5) it is well known that (4) has a unique positive steadystate {ui }i∈Ω∗ , which satisfies{
ui (mi − ui ) + δ(ui−1 + ui+1 − 2ui ) = 0,i ∈ Ω∗ ,(6)
u0 = u1 , uN +1 = uN .
Furthermore, the unique positive steady state is globally stable and the total population of (4) satisfies
N∑
i=1

vi (t) →

N
∑

ui

i=1

7

as t → ∞.

∑N
Our purpose is to maximize the total population U :=i=1 ui at equilibrium
under the constraint (5). See [7, 9, 12, 24] for related works.
This sort of multi-patch model is called “island chain” model or “steppingstone” model. Such model views the space as a collection of discrete patches. We
treat each patch as a point, and view the overall population of a single species
as a vector, with each component corresponding to the number of individuals ineach patch. Furthermore, we can treat the dispersal in this model as a discrete
analogue of the continuous diﬀusion. For more details see [1, 5, 8] and references
therein. For this reason, this work is closely relevant to the investigation of thereaction-diﬀusion equation (2) in one-dimensional case. We also refer to [5, 15]
and references therein for previous works of (2).
The maximization of the total population for the steady state of (2) has recently
been studied by [16, 21] in Ω ⊂ Rd . They showed under some conditions that any
global maximizer of the total population for the steady state must be of “bangbang” type, which gives a partial answer to the conjecture raised by Ding et al.
[6]. More recently, Mazari et al. [16] proved that if δ > 0 is suﬃciently large,
then the global maximizer is given by m(x) := χE , where either E = (0, m)
or (1 − m, 1). Their numerical simulation results indicated that if the diﬀusion
constant is suﬃciently small, then fragmentation may occur in the one-dimensionalcase. However, it is extremely diﬃcult to explicitly determine the maximizer forthe steady state of (2) in general.
This motivates us to study the maximization problem for the diﬀerence equation (6), for which the computations of the total population can be done (butstill fairly non-trivial) for small and large diﬀusion rates. Our results show thatthe global maximizer depends crucially on the diﬀusion rate δ, and the answers
are completely diﬀerent for small δ and large δ. In several cases we are able toshow that the global maximizer is of the “bang-bang” type and to determine themaximizers explicitly by finding the specific guiding rules of fragmentation in themulti-patch model (6). In particular, fragmentation occurs when the diﬀusion rateis suﬃciently small, which echoes the numerical findings in [16]. In this paper, we
do not assume the upper bound for the resource distribution in each patch.
We define the set
M := {{mi }i∈Ω∗ | {mi }i∈Ω∗ satisfies (5)} ⊂ RN .
For convenience, we express {mi }i∈Ω∗ by m or (m1 , m2 , . . . , mN ). Note that thesolution of (6) depends on the diﬀusion constant δ > 0 and resources m ∈ M. We
denote the total population at stable equilibrium as U = U (m, δ).
Our first main result is stated as follows:
Theorem 3 (Global maximizer for large δ). Suppose that m ∈ M is given bym = (0, 0, 0, . . . , m) or m = (m, 0, 0, . . . , 0).
8

Then there exists a positive constant ∆N,m > 0 such that for any δ > ∆N,m andfor any m̃ ∈ M \ {m}, the total population satisfies U (m, δ) > U (m̃, δ).
Note that this theorem is consistent with the result of [16] for (2).
The next theorem shows that the global maximizer for suﬃciently small δ isfragmented, and there are some rules of fragmentation in the multi-patch model(6).
Theorem 4 (Global maximizer for small δ). Let p > 0 and r > 0 be any two positive integers. Define Pm = (0, m/p, 0), Pm∗ = (0, m∗ , 0), m∗ = (0, m∗ , 0, m∗ , 0),
where√(1 + 2)2 m
4m
∗
√
√ .
m =
, m∗ =22{4(p − 1) + (1 + 2) }4(p − 1) + (1 + 2)2
Choose η ∈ (0, p∗ ] arbitrarily, where
{
m/p
p∗ :=
m∗

if N = 3p, 3p + 1,
if N = 3p + 2.

Define a set
Mη := {m ∈ M | mi ≥ η or mi = 0 for all i ∈ Ω∗ }.
Suppose that m ∈ Mη is given as follows:(i) If N = 3p, thenm = (Pm , Pm , . . . , Pm ).
{z
}
|
p

(ii) If N = 3p + 1, there are 3 cases:(a) For p = 1, thenm = (0, Pm ), or m = (Pm , 0).
(b) For p = 2r, then
3r+1

m = (Pm , . . . , Pm , 0̌ , Pm , . . . , Pm ).
|
{z
}
|
{z
}
r

r

(c) For p = 2r + 1, then
3r+1

m = (Pm , . . . , Pm , 0̌ , Pm , . . . , Pm ),
{z
}
{z
}
|
|
r

r+1

9

or
3(r+1)+1

m = (Pm , . . . , Pm ,
|
{z
}

0̌

, Pm , . . . , Pm ).
{z
}
|

r+1

r

(iii) If N = 3p + 2, there are 3 cases:(a) For p = 1, then( mm )m = 0, , 0, , 0 .
22(b) For p = 2r, thenm = (Pm∗ , . . . , Pm∗ , m∗ , Pm∗ , . . . , Pm∗ ),
{z
}
{z
}
|
|
r−1

or

r

m = (Pm∗ , . . . , Pm∗ , m∗ , Pm∗ , . . . , Pm∗ ).
{z
}
{z
}
|
|
r

r−1

(c) For p = 2r + 1, thenm = (Pm∗ , . . . , Pm∗ , m∗ , Pm∗ , . . . , Pm∗ ).
|
|
{z
}
{z
}
r

r

Then there exists positive constant δN,m,η > 0 such that for any δ ∈ (0, δN,m,η ) andany m̃ ∈ Mη \ {m}, it follows that U (m, δ) > U (m̃, δ).
Remark 5. Let m ∈ M be given as in Theorem 4. Choose m̃ ∈ M \ {m}arbitrarily. The proof of theorem 4 implies that there exists positive number δm̃such that for any δ ∈ (0, δm̃ ), it follows that U (m, δ) > U (m̃, δ). Hence m̃ ∈M \ {m} is not global maximizer for suﬃciently small δ.
This theorem refers to a global-maximizer when there is minimum amount of
resources η > 0 for each patches. When η = 0, it is very diﬃcult to determine a
global maximizer. However, in the two patches case, we obtain global maximizers
for all δ > 0.
Theorem 6 (Global maximizer for two patches). In the case N = 2, Suppose thatm ∈ M is given as
m = (0, m), or m = (m, 0).
Then for any m̃ ∈ M \ {m}, it follows that U (m, δ) > U (m̃, δ) for all δ > 0.
Remark 7 (Global minimizer). It is known that a global minimizer of the total
population of (2) is constant. It still holds when we use the multi-patch model, thatis, mi ≡ m/N for all i ∈ Ω∗ is the global minimizer for all δ > 0.
We conclude this summary by noting that the author’s Ph.D thesis [19] is basedon [20, 21].
10

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